Among its many interpretations, the term reliability most commonly refers to the ability of a device or system to perform a task successfully when required. More formally, it is described as the probability of functioning properly at a given time and under specified operating conditions [1]. Mathematically, the reliability function is defined by

where is a nonnegative random variable representing the device or system lifetime.

For a system composed of at least two components, the system reliability is determined by the reliability of the individual components and the relationships among them. These relationships can be depicted using a reliability block diagram (RBD).

Simple systems are usually represented by RBDs with components in either a series or parallel configuration. In a series system, all components must function satisfactorily in order for the system to operate. For a parallel system to operate, at least one component must function correctly. Systems can also contain components arranged in both series and parallel configurations. If an RBD cannot be reduced to a series, parallel, or series-parallel configuration, then it is considered a complex system.

This article deals with the generation of an exact analytical expression for the reliability of a complex system. The demonstrated method relies on finding all paths between the source and target vertices in a directed acyclic graph (i.e., RBD), as well as the inclusion-exclusion principle for probability.

**A Note on Timings**

The timings reported in this article were measured on a custom workstation PC using the built-in function `Timing`. The system consists of an Intel® Core i7 CPU 950 @ 4 GHz and 24 GB of DDR3 memory. It runs Microsoft® Windows 7 Professional (64-bit) and scores 1.32 on the *MathematicaMark9* benchmark.

We begin by considering a directed graph that consists of a finite set of vertices together with a finite set of ordered pairs of vertices called directed edges. The built-in function `Graph` can be used to construct a graph from explicit lists of vertices and edges.

This two-dimensional grid graph, labeled , can be constructed much more efficiently by using the built-in function `GridGraph`. Throughout this section, we utilize it to illustrate our functions.

Now, for a vertex , we define the set of out-neighbors as

where is taken to mean a directed edge from to . This is implemented in the function `VertexOutNeighbors`.

`VertexOutNeighbors` behaves similarly to the built-in function `VertexOutDegree`. That is, given a graph and a vertex , the function returns a list of out-neighbors for the specified vertex.

If, however, only the graph is specified, the function will give a list of vertex out-neighbors for all vertices in the graph.

The order in which the out-neighbors are displayed is determined by the order of vertices returned by `VertexList`.

We can implement similar functions to obtain the set of in-neighbors by simply changing to .

The next step toward our goal is to consider a method of traversing a graph. One common approach of systematically visiting all vertices of a graph is known as depth-first search (DFS). In its most basic form, a DFS algorithm involves visiting a vertex, marking it as “visited,” and then recursively visiting all of its neighbors [2]. The function `DepthFirstSearch` implements this algorithm for directed graphs.

Given a graph and a starting vertex , `DepthFirstSearch` returns a list of vertices in the order in which they are visited.

We compare this with the result of the built-in function `DepthFirstScan`.

Next, let us define the function `DirectedAcyclicGraphQ`.

If the graph is both directed and acyclic, `DirectedAcyclicGraphQ` yields `True`. Otherwise, it yields `False`.

Finally, we consider the problem of finding all paths in a directed acyclic graph between two arbitrary vertices . Typically, we refer to as the source and as the target. A path in is defined as a sequence of vertices such that for . Since we have constrained ourselves to a directed acyclic graph, all paths are simple. That is to say, all vertices in a path are distinct.

By modifying the depth-first search algorithm, we arrive at a solution.

Like the original DFS algorithm, we visit a vertex and then recursively visit all of its neighbors. However, instead of checking if a vertex has been marked “visited,” we compare the current vertex to the target. If they do not match, we continue to traverse the graph. Otherwise, the target has been reached and we store the path for later output.

For a given directed acyclic graph , a source vertex , and a target vertex , `FindPaths` returns a list of all paths connecting to .

In this particular instance, the function takes approximately 0.85 milliseconds to return the result.

`FindPaths` works for any pair of vertices.

If no path is found, the function returns an empty list.

Up to this point, we have been working with graphs in an abstract, mathematical sense. We now make the transition from directed acyclic graph to reliability block diagram by associating vertices with components in a system and edges with relationships among them.

Consider a single component in an RBD. Let us imagine a “flow” moving from a source, through the component, to a target. The component is deemed to be functioning if the flow can pass through it unimpeded. However, if the component has failed, the flow is prevented from reaching the target.

The “flow” concept can be extended to an entire system. A system is considered to be functioning if there exists a set of functioning components that permits the flow to move from source to target. We define a path in an RBD as a set of functioning components that guarantees a functioning system. Since we have chosen to use a directed acyclic graph to represent a system’s RBD, all paths are minimal. That is to say, all components in a path are distinct.

Once the minimal paths of a system’s RBD have been obtained, the principle of inclusion-exclusion for probability can be employed to generate an exact analytical expression for reliability. Let be the set of all minimal paths of a system. At least one minimal path must function in order for the system to function. We can write the reliability of the system as the probability of the union of all minimal paths:

This is implemented in the function `SystemReliability`.

Given a system’s RBD (represented by a directed acyclic graph ), a source vertex , and a target vertex , `SystemReliability` returns an exact analytical expression for the reliability.

Consider the RBD of a simple system with four components in a series configuration.

The reliability of the system is given in terms of the reliability of its four components.

Consider the RBD of a simple system with four components in a parallel configuration.

The “start” and “end” components are not part of the actual system. They are added to ensure the RBD meets the criteria for a directed acyclic graph.

Furthermore, these nonphysical components are taken to have perfect reliability, that is, . Since they have no effect on the system’s reliability, they can be safely removed from the resulting analytical expression. To do so, we simply define a list of replacement rules and apply it to the result of `SystemReliability`.

The reliability of the system is given in terms of the reliability of its four components.

Next, we examine the RBDs of two simple systems with components in a series-parallel configuration.

Component is in series with component , and both components are in parallel with component .

As in previous examples, we use `SystemReliability` to obtain an exact analytical expression for the reliability.

Finally, we examine the RBDs of two complex systems.

The reliability of the system is given in terms of the reliability of its six components.

The result is returned after approximately 0.59 milliseconds.

The reliability of the system is given in terms of the reliability of its fourteen components.

The result is returned after approximately 0.33 seconds.

We now turn our attention to the derivation of a time-dependent expression for the reliability of a complex system based on information contained within its reliability block diagram.

Let us imagine that we have a generic system composed of six subsystems and we know the reliability relationships among them. In addition, the underlying statistical distributions and parameters used to model the subsystems’ reliabilities are known.

We begin by creating the system’s RBD.

In defining the RBD, we have made use of the `Property` function to store information associated with each subsystem. For instance, the custom property `"Distribution"` is used to store a parametric statistical distribution. Labels, images, and other properties can also be specified.

Next, we use `SystemReliability` to generate an exact analytical expression for the reliability.

Now, the reliability function of the subsystem is given by

where is the corresponding cumulative distribution function (CDF). For each subsystem, we use `PropertyValue` to extract the symbolic distribution stored in the RBD, and then use the built-in function `CDF` to construct its reliability function.

We extract additional information, for example, subsystem labels, from the RBD and combine it with the reliability functions to create plots for comparison.

In order to transform our static analytical expression into a time-dependent function, we first define a list of replacement rules.

Next, we apply the list of rules to the expression for system reliability.

The result is a time-dependent reliability function for the complex system described by the RBD.

Finally, we generate a plot of the system’s reliability over time.

We have demonstrated a method of generating an exact analytical expression for the reliability of a complex system using a directed acyclic graph to represent the system’s reliability block diagram. In addition, we have shown how to convert an analytical expression for system reliability into a time-dependent function based on statistical information stored in an RBD. While our focus has been on the analysis of complex systems, we have also shown that the combination of path finding and the inclusion-exclusion principle is equally applicable to simple systems in series, parallel, or series-parallel configurations.

Knowing the static analytical expression or time-dependent solution of a system allows us to perform a more advanced reliability analysis. For instance, we can easily calculate the Birnbaum importance

of the component using the result of `SystemReliability`. Similarly, we can derive the hazard function, or failure rate, from the system’s time-dependent reliability function.

There are several ways in which the functionality demonstrated in this article can be improved and expanded:

- Increase the efficiency of
`SystemReliability`by implementing improvements to the classical inclusion-exclusion principle [3]. - Add functions related to common tasks in reliability analysis, for example, reliability importance, failure rate, and so on.
- Add support for -out-of- structures, that is, redundancy.
- Add the ability to export and import complete RBDs.
- Add a mechanism, for example, a graphical user interface (GUI), to facilitate the construction and modification of RBDs.

Finally, the code can be combined into a user-friendly package with full documentation.

[1] | W. Kuo and M. Zuo, Optimal Reliability Modeling: Principles and Applications, Hoboken, NJ: John Wiley & Sons, 2003. |

[2] | S. Skiena, The Algorithm Design Manual, 2nd ed., London, UK: Springer-Verlag, 2008. |

[3] | K. Dohmen, “Improved Inclusion-Exclusion Identities and Inequalities Based on a Particular Class of Abstract Tubes,” Electronic Journal of Probability, 4, 1999 pp. 1-12. doi:10.1214/EJP.v4-42. |

T. Silvestri, “Complex System Reliability,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-7. |

Todd Silvestri received his undergraduate degrees in physics and mathematics from the University of Chicago in 2001. As a graduate student, he worked briefly at the Thomas Jefferson National Accelerator Facility (TJNAF) where he helped to construct and test a neutron detector used in experiments to measure the neutron electric form factor at high momentum transfer. From 2006 to 2011, he worked as a physicist at the US Army Armament Research, Development and Engineering Center (ARDEC). During his time there, he cofounded and served as principal investigator of a small laboratory focused on improving the reliability of military systems. He is currently working on several personal projects.

**Todd Silvestri**

*New Jersey, United States*

*todd.silvestri@optimum.net*

The aim of canonical correlation analysis is to find the best linear combination between two multivariate datasets that maximizes the correlation coefficient between them. This is particularly useful to determine the relationship between *criterion measures* and the set of their *explanatory factors*. This technique involves, first, the reduction of the dimensions of the two multivariate datasets by projection, and second, the calculation of the relationship (measured by the correlation coefficient) between the two projections of the datasets.

While the correlation coefficient measures the relationship between two simple variables, canonical correlation analysis measures the relationship between two *sets* of variables. Although the correlation measure employed for both techniques is the same, namely

(1) |

the distinction between the two techniques must be clear: while for the correlation coefficient and must be -dimensional vectors containing realizations of the random variables, for canonical correlation analysis (CCA) has to be an and an matrix, with and at least 2. In the latter case, is the number of realizations for all random variables, where is the number of random variables contained in the set and is the number of random variables in the set .

This article calculates, through CCA, the relationship between stock markets of developed and developing countries and performs Bartlett’s test for the statistical significance of the canonical correlation found.

For an introduction to statistics in financial markets, see [1].

The data employed for the CCA in the present work was obtained directly from *Mathematica*’s function. The variables are divided into two groups: the ETFs representing developed nations and the ETFs representing developing countries. The first group is treated as independent variables and the second group as dependent variables. The idea here is to analyze the relationship between stock markets in these two groups of countries through ETFs traded at the New York Stock Exchange (NYSE).

Although there are several country-specific ETFs traded on the NYSE, not all of them were chosen. The idea is to select, for each group, those ETFs representing countries with large stock markets according to a market capitalization criterion. The market capitalization of all stock markets was obtained from the website of the World Federation of Exchanges (www.world-exchanges.org/statistics). All countries with stock markets greater than 500 billion US dollars in December 2012 were chosen, and only one ETF per country was selected.

These six ETFs were included in the group of developed nations: EWA (Australia), EWC (Canada), EWG (Germany), EWJ (Japan), EWU (UK), and SPY (USA).

Eight ETFs were included in the group of developing countries: EWZ (Brazil), FXI (China), EPI (India), EWW (Mexico), RSX (Russia), EWS (Singapore), EWY (South Korea), and EWT (Taiwan).

These are the monthly returns for the five-year period between March 2008 and February 2013 (60 months).

This checks the number of observations for each variable. Evaluate the previous command again if the lengths are not all 60.

This plots the data for all the variables.

This plots the price behavior of the six ETFs representing developed countries for the 60-month period.

This plots the price behavior of the eight ETFs representing developing countries for the 60-month period.

According to [2], “to use canonical correlation analysis safely for descriptive purposes requires no distributional assumptions.” However, they still state that “to test the significance of the relationships between canonical variates, (…), the data should meet the requirements of multivariate normality and homogeneity of variance” ([2], p. 339). Is the data normally distributed in this sense?

As can be seen, the null hypothesis of normality cannot be rejected for all variables at the 5% confidence level.

In order to perform the canonical correlation analysis, it is necessary to organize the data into two groups of variables: (representing the developed countries) and (representing the developing countries);

where to represent the developed countries’ ETFs and to represent the developing countries’ ETFs.

In canonical correlation analysis, and , and the problem is to find the “most interesting” linear combinations

for the two sets of variables, that is, those values that maximize

(2) |

Let be the concatenation of the matrices and ,

so

where and are the (empirical) variance-covariance matrices and and are the mean vectors of and , respectively. represents the covariance matrix of and , and is its transpose.

From equation (1) and from the properties

(3) |

(4) |

where and are conformable and is a constant,

(5) |

where

CCA can be performed either on variance-covariance matrices or on correlation matrices. If the random variables and are standardized to have unit variance, the variance-covariance matrix becomes a correlation matrix.

After partitioning the variance-covariance matrix, and given equation (5), the main objective is to solve

(6) |

subject to

To solve this problem, define:

(7) |

A singular value decomposition of gives

(8) |

where

(9) |

(10) |

(11) |

and are column orthonormal matrices , and is a diagonal matrix with positive elements, namely, the eigenvalues of . (For detailed information about singular value decomposition, see [3].) From the property

and from equation (7),

For this solution procedure, the largest eigenvalue of is the canonical correlation of our analysis. and can also be found through

(12) |

(13) |

The problem in this case is to solve the following canonical equations [2, 4]:

(14) |

and

(15) |

where is the identity matrix and is the largest eigenvalue for the characteristic equations

(16) |

and

(17) |

The largest eigenvalue of the product matrices

is the squared canonical correlation coefficient. Furthermore, it can be shown that

(18) |

and

(19) |

which means that only one of the characteristic equations needs to be solved in order to find or .

This transposes the data.

This checks the dimensions of `Z`; it has 60 rows (months) and 14 columns (ETFs).

There are 14 random variables (six in the first set and eight in the second); the dimensions of the submatrices are 60×6 for , 60×8 for , 6×6 for , 6×8 for , 8×6 for , and 8×8 for .

Define `M1` to be the variance-covariance matrix of `Z`. Here are the first seven columns of `M1`.

Partition `M1` into the four submatrices , , , and .

To better understand the relationship between the random variables, here is `M2`, the correlation matrix of `Z`.

This defines `K`.

This performs the singular value decomposition on `K`.

This is the largest eigenvalue of `K`.

This checks by computing the square root of the eigenvalues of

and

according to the second solution procedure. (`Chop` replaces numbers that are close to zero by the exact integer 0.)

Performing a spectral decomposition on and and calculating the square roots of their eigenvalues is another check of the canonical correlation coefficient.

The checks agree.

The last step in this analysis is to find the canonical correlation vectors, which maximize the correlation between the canonical variates. According to equations (12) and (13), this computes the canonical correlation vectors.

The canonical correlation matrix ` B` is computed using , not , because

Given that

the canonical correlation vectors and are the columns of and .

In terms of the canonical correlation vectors, the canonical variates are

where, as before,

Given that

(20) |

only and are needed in order to find . Thus, the only canonical variates needed are and .

The interpretation of canonical correlation coefficients, canonical correlation vectors, and canonical variates is one of the most difficult tasks in the whole analysis. CCA would be better understood relating the original data matrix to the matrix computed using the canonical correlation vectors, which is simply a reduction of the data matrix through linear combinations of its elements. It should be easier to understand that the canonical correlation coefficient is merely the ordinary Bravais-Pearson correlation between the two columns of the reduced matrix.

In principle, one can say that the highest canonical correlation coefficient that was found is the maximum possible correlation between the two columns of the reduced matrix. In this case, it is usual to say that this coefficient represents the relationship between the two datasets, and , in the sense of a correlation measure. Thus, if is the matrix containing the explanatory factors of , the matrix containing the criterion measures (or criterion variables), it is possible to say that the explanatory factors would perfectly explain the criterion variables if . If , the explanatory factors have no influence on the criterion variables, and any value between 1 and 0 is merely an interpolation of these extreme cases.

In the next inputs we will compute and show (partially) the reduced data matrix. In order to demonstrate the validity of the CCA theory, we also compute the correlation for the other (not so interesting for our analysis) canonical variates. We start by defining and .

The first column of our reduced data matrix is .

The first value of , for instance, refers to the linear combination between EWA, EWC, EWG, EWJ, EWU, and SPY for March 2008, such that

We can also define .

Thus, after assigning the values to the canonical variates, , , , and , we have four vectors with the values of the linear combinations of and . Now we can simply compute the Bravais-Pearson correlation between all the canonical variables.

We also verify equation (20).

The correlation between the canonical variates can be better interpreted graphically. First we show the reduced matrix computed using the canonical correlation vectors and , whose canonical correlation coefficient is .

Now we show the reduced matrix computed using the canonical correlation vectors and , whose canonical correlation coefficient is .

Finally, we compute the *canonical loadings*, that is, the correlation between every single ETF and its respective canonical variate.

We can also compute the *canonical cross-loadings*, that is, the correlation between every single ETF and its opposite canonical variate.

It might be of interest to compute the canonical loadings for the *second canonical variate*, that is, the linear combination of variables with correlation coefficient .

Finally, we compute the canonical cross-loadings for the second canonical variate, that is, the linear combination of variables with correlation coefficient .

It is possible to compute canonical loadings and cross-loadings for all the six canonical variates. However, only the first two are shown here for descriptive purposes.

In this section we test the hypothesis of no correlation between the two sets and . An approximation for large was provided in [5]:

(21) |

where

We can also test the hypothesis that the individual canonical correlation coefficients are different from zero:

(22) |

where is a parameter to select the canonical correlation coefficient to be tested.

This defines the Bartlett variable.

This assigns values to the .

We calculate Bartlett’s statistic (equation (21)) to test if the two sets of variables and are uncorrelated. Our hypotheses are:

This computes the 99% quantile of the chi-square distribution with 48 () degrees of freedom, .

**Test Conclusion**: The hypothesis of no correlation between the two sets has to be rejected once the Bartlett statistic (here 249.415) is greater than the 99% quantile of the chi-square distribution with 48 degrees of freedom (here 73.6826).

This article analyzed the relationship between two sets of variables, namely financial assets represented by NYSE-traded country-specific ETFs. The ETFs were divided into two sets representing developed and developing countries. In the first set a total of six ETFs (representing developed countries) were included, while in the second set a total of eight ETFs were included (representing developing countries). Using monthly return data for a five-year period it was possible to show, through canonical correlation analysis (CCA), that there is a significant relationship between these two sets of ETFs. The highest correlation coefficient found in the present study was and, in an analogous manner to statistics in regression analysis, we could interpret its squared value as the explanatory power of the canonical correlation analysis. In other words, the squared canonical correlation coefficient indicates the proportion of variance a dependent variable linearly shares with the independent variable generated from the observed variable’s set (i.e., the canonical variates).

[1] | J. Franke, W. Härdle, and C. Hafner, Einführung in die Statistik der Finanzmärkte, Berlin: Springer Verlag, 2001. |

[2] | W. R. Dillon and M. Goldstein, Chap. 9 in Multivariate Analysis: Methods and Applications, New York: Wiley, 1984. |

[3] | K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis, London: Academic Press, 1979. |

[4] | T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd ed., New York: Wiley, 2003. |

[5] | M. S. Bartlett, “A Note on Tests of Significance in Multivariate Analysis,” Proceedings of the Cambridge Philosophical Society, 35(2), pp. 180-185, 1939. doi:10.1017/S0305004100020880. |

R. L. Malacarne, “Canonical Correlation Analysis,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-6. |

Rodrigo Loureiro Malacarne is a professor of financial mathematics and financial management at the Faculdades Integradas Espirito Santenses (FAESA). His areas of research include statistics of financial markets and financial time series analysis.

**Rodrigo Loureiro Malacarne
**

Faculdades Integradas Espirito Santenses (FAESA)

Av. Vitória, 2.220 – Monte Belo

Vitória, ES, Brazil – CEP 29.053-360

malacarne@gmail.com

Motivated by the computational advantages offered by *Mathematica,* I decided some time ago to embark on collecting and implementing properties of the fascinating geometric figure called the arbelos. I have since been impressed by the large number of surprising discoveries and computational challenges that have sprung out of the growing literature concerning this remarkable object. I recall its resemblance to the lower part of the iconic canopied penny-farthing bicycle of the 1960s TV series *The Prisoner*, Punch’s jester cap (of *Punch and Judy* fame), and a yin-yang symbol with one arc inverted; see Figure 1. There is now an online specialized catalog of Archimedean circles (circles contained in the arbelos) [1] and important applications outside the realm of mathematics and computer science [2] of arbelos-related properties.

Many famous names are involved in this fascinating theme, among them Archimedes (killed by a Roman soldier in 212 BC), Pappus (320 AD), Christian O. Mohr (1835-1918), Victor Thébault (1882-1960), Leon Bankoff (1908-1997), and Martin Gardner (1914-2010). Recently, they have been succeeded by Clayton Dodge, Peter Y. Woo, Thomas Schoch, Hiroshi Okumura, and Masayuki Watanabe, among others.

Leon Bankoff was the person who stimulated the extraordinary attention on the arbelos over the last 30 years. Schoch drew Bankoff’s attention to the arbelos in 1979 by discovering several new Archimedean circles. He sent a 20-page handwritten note to Martin Gardner, who forwarded it to Bankoff, who then gave a 10-chapter manuscript copy to Dodge in 1996. Due to Bankoff’s death, a planned joint work was interrupted until Dodge reported some discoveries [3]. In 1999 Dodge said that it would take him five to ten years to sort all the material in his possession, then filling three suitcases. Currently this work is still forthcoming. Not surprisingly, like Volume 4 of *The Art of Computer Programming*, it appears that important work needs a substantial time to be developed.

**Figure 1.** *The Prisoner’*s penny-farthing bicycle, Punch and Judy, a physical arbelos.

The arbelos (“shoemaker’s knife” in Greek) is named for its resemblance to the blade of a knife used by cobblers (Figure 1). The arbelos is a plane region bounded by three semicircles sharing a common baseline (Figure 2). Archimedes appears to have been the first to study its mathematical properties, which he included in propositions 4 through 8 of his *Liber assumptorum* (or* Book of Lemmas*). This work might not be entirely by Archimedes, as was recently revealed through an Arabic translation of the *Book of Lemmas* that mentions Archimedes repeatedly without fully recognizing his authorship (some even believe this work to be spurious [4]). The *Book of Lemmas* also contained Archimedes’s famous *Problema Bovinum* [5].

This article aims at systematically enumerating selected properties of the arbelos, without attempting to be exhaustive. Our purpose is to develop a uniform computational methodology in order to tackle those properties in a pedagogical setting. A sequence of properties is arranged and subsequently verified by testing the computationally equivalent predicates. This work includes some discoveries and extensions contributed by the author.

We refer to the largest semicircle as the *top arc* and the two small ones as the left and right *side* *arcs,* or just the *side* *arcs* when there is no need to distinguish them. We use and to denote their respective radii (the top arc thus has radius ). A *segment* between two points is an undirected line segment going from one point to the other, while a *line* through two points is the infinite straight line through the two points. A traditional abuse of notation uses for both the line segment joining the points and and the length of the segment, depending on the context; modern usage is to write for the length of the segment.

This function displays the arbelos.

This draws the basic arbelos.

**Figure 2.** The arbelos.

**Property 1**

In other words, the total length of the side arcs equals the length of the top arc. This property is related to an intriguing paradox [6].

**Property 2**

This was lemma 4 of the *Book of Lemmas *(see Figure 3) [7, 8].

These two properties are easily verified by simultaneously testing two equalities.

The function `drawpoints` is used to display specific points as red disks.

**Figure 3.** The area of the circle of diameter (the radical circle) is equal to the area of the arbelos.

The circle in Figure 3 is called the *radical circle* of the arbelos and the line is its *radical axis* (this terminology will be clarified in Generalizations). To illustrate properties 3-11 and 25, 26, we draw and label points and show some coordinates, lines, and circles in Figure 4.

**Figure 4.** Labels, coordinates, lines, and circles referred to in properties 3 through 11 and 25, 26.

**Property 3**

The lines and are orthogonal and intersect the side arcs at points and , joining a common tangent to the side arcs.

To verify the orthogonality of the lines and , we take the inner product of the vectors and .

We employ the following result to obtain the slopes at the points and .

**Theorem 1**

The function `PQ` finds the coordinates of the tangent points and by solving a system of four equations, which places them on the arcs and sets their tangent slopes according to theorem 1.

Besides `PQ`, other definitions in this article for points and quantities are: `VWS`, `HK`, `U`, `EF`, `IJr`, and `LM`.

The function `dSq` computes the square of the distance between two given points.

**Property 4**

As is a diameter of the radical circle, we only need to verify the equality of the distances of and to the center of the radical circle, namely the point .

**Property 5**

Let the line intersect the top arc at points and . Then and lie on a circle with center and radius .

We get the coordinates of the points and by solving a system of equations that places them on the top arc and on the line .

This verifies property 5 by checking that the distances of and to are the same as the distance from to .

**Property 6**

This is equivalent to the fact that the determinant (cross product) of the vectors and is zero.

**Property 7**

This is equivalent to the fact that the inner product of the vectors and is zero.

Let us use the notation for a circle with center and radius .

**Property 8**

The inversion of a point in the circle , is defined to be the unique point such that [9]. The function `inversion` implements this idea.

This verifies property 8, recalling the coordinates of are .

**Property 9**

Let be the circle of inversion. The points , , invert to themselves. The segment inverts to the arc and the segment inverts to the arc . The arcs and invert to themselves. The radical circle inverts to the line .

**Property 10**

This is the same as claiming that the corresponding arcs are orthogonal to the radical circle. By property 8, the arcs are orthogonal to the circle with diameter as they pass through inverse pairs [10, 11].

**Property 11**

This is one of Bankoff’s surprises [12, 13, 14]. As all four points are on the radical circle, we need to verify only that bisects .

The following `Manipulate` illustrates properties 3-11. The easiest way to define the points `P`, `Q`, `H`, `K` is to copy and paste the formulas for them.

Now consider the circle tangent to the side arcs and the top arc, the *incircle* with tangent points , , and as shown in Figure 5 [15, 16]. We also consider points and at the tops of the side arcs.

**Figure 5.** The incircle and coordinates, lines, and points referred to in properties 12 through 15.

Proposition 6 of the *Book of Lemmas* included the value of , the radius of the incircle. The function `U` calculates the coordinates of the center and the radius .

The coordinates of the tangent points , , and are obtained as the intersections of the lines joining the centers of the three arcs of the arbelos and the incircle.

**Property 12**

The points , , and are collinear. The points , , and are collinear. The lines and intersect in a point lying on the incircle.

Using the criterion of the determinant to check for collinearity, we verify the first two claims.

Let be the point of intersection of the lines and . Confirming that its distance to is equal to verifies the third claim.

**Property 13**

The points , , , and are on a circle with center . Similarly, the points , , , and are on a circle with center .

The following `Manipulate` illustrates property 13 [17]. The option for showing the Bankoff circle as the incircle of the triangle joining the center of the arcs and the incircle corresponds to property 23.

**Property 14**

Let be the diameter of the incircle parallel to and let be the projection of onto . The rectangle between the segments and is a square.

This property is illustrated in the next `Manipulate` and is readily verified here.

**Property 15**

Let and be the intersections of the lines and with the side arcs. Then is a square of almost the same size as the one mentioned in property 14.

First we obtain points and as the intersections of their respective lines and their respective arcs, and keep the result in the variable `replaceEF`.

We verify property 15 by setting to be equal to the vector obtained by rotating around by 90° and setting to be equal to the vector obtained by translating by .

Assuming and, the following plot compares the sizes of the two squares.

This `Manipulate` illustrates properties 14 and 15.

Consider the two gray circles tangent to the radical axis, a side arc, and the top arc in Figure 6. They are called *the twins*, or the *Archimedean circles*. Due to the following remarkable property, they have been extensively studied. We collect many of their extraordinary occurrences in our list of properties [3, 18, 19].

**Figure 6.** The twins.

**Property 16**

The two circles tangent to the radical axis, the top arc, and one of the side arcs of an arbelos have the same radius.

This property appeared as proposition 5 in the *Book of Lemmas*. Solving the following system of six equations finds the values of the radii, verifies they are equal, and computes the centers , .

These four solutions give the centers in pairs: , , , , where and are the reflections of and in the diameter of the arbelos; only the last expression is valid. The result also shows that the twins are indeed of the same radius . Any circle with radius equal to the twins’ radius is called *Archimedean*. A nice interpretation of arises when considering and as resistances: then is the resistance resulting from connecting and in parallel; that is, . The function `IJr` computes the value of the centers and the common value of the radius of the twins.

**Property 17**

Consider a circle tangent to both twins, with center at point and radius . Then there are two possible values of .

To find the extrema of , we set the derivative of each of the above expressions to zero and solve for .

So the centers of the smallest and largest circles tangent to the twins lie on the radical axis. Moreover, they are concentric, as this result confirms.

Thus, by using property 2, we confirm that the largest tangent circle, which is the smallest enclosing the twins, satisfies property 17. The following `Manipulate` shows the circles tangent to the twins as you vary the radius of the left side arc.

The following plot compares the radii of the two circles tangent to the twins with centers on the radical axis.

**Figure 7.** Labels and lines referred to in properties 18 through 24.

**Property 18**

The common tangent of the left arc and its tangent twin at passes through . Similarly, the common tangent of the right arc and its tangent twin at passes through (see Figure 7).

This computes the tangent points and .

By using theorem 1, we verify both claims.

**Property 19**

We verify both claims simultaneously.

However, the points , , and are not on a circle centered at , nor are the points , , and on a circle centered at ; otherwise, the following expression would be zero.

**Property 20**

As the length of the segment is the ordinate of and the length of the segment is the ordinate of , we only need to verify that the midpoints of those segments lie on the mentioned lines by checking slopes.

**Property 21**

Those circles are the fourth and fifth Archimedean circles discovered by Bankoff [20]. In order to verify this property, we use the following result [21]:

**Theorem 2**

This directed distance is positive if the triangle is traversed counterclockwise and negative otherwise. The function `dAB` implements this.

Let and be the center and radius of the blue circle on the left side of point in Figure 7. Solving the following system finds the value of .

Similarly, this calculates the radius of the blue circle to the right of , which equals .

Thus, both circles are Archimedean as claimed. The following `Manipulate` shows the twins and these two other circles.

**Property 22**

Archimedes discovered the original twins; Bankoff improved on this by discovering this third circle in 1950 [22]. The coordinates of the center of the Bankoff circle are obtained by equating the distances of to the points , , and .

**Property 23**

The Bankoff circle is the incircle of the triangle formed by joining the centers of the side arcs and the center of the incircle of the arbelos.

Using theorem 2 to compute the distance of to the sides of the triangle, we verify this property (as `dAB` computes a directed distance, the order of the arguments describing the line is important).

**Property 24**

This computes the values of and .

The circle is the one where the ordinate of is positive. Note that is not on the radical axis.

**Property 25**

The circles and tangent to the radical axis, one passing through and the other passing through the point , are both Archimedean (see Figure 4).

**Property 26**

A circle with center and radius tangent to the line is such that the distance from to is

, so this equation holds:

Because the circle passes through ,

Because the circle is tangent to the top arc,

This input uses explicit expressions for , , and that satisfy these three equations.

**Property 27**

Consider the two (red) segments connecting the center of the top arc to the top points and of the left and right arcs of the arbelos. These segments have the same length and are orthogonal. The tangent circles and at and to those lines and the top arc are Archimedean (see Figure 8).

This property was discovered in the summer of 1998 [23].

**Figure 8.** The two pairs of Archimedean circles from property 27.

We have seen that there are some Archimedean circles other than the twins, namely the Bankoff circle and those mentioned in properties 21 through 27. There are also *non-Archimedean twins*, that is, pairs of circles of the same radius, different than that of the twins, appearing at significant places within the arbelos.

The discovery of the *slanted twins *arose from the initial assumption that, besides being tangent to either side arc and the top arc, the two circles-to-be-twins could be tangent to themselves and not necessarily to the radical axis. Clearly there are an infinite number of solutions if we do not require these circles to be of equal radius. The idea was that if we started by assuming they are of equal radius, we might end up discovering they are tangent to the radical axis. This turned out not to be the case. Let us consider circles with centers at the points and with common radius . The value of can be obtained by solving a system of five equations.

These expressions involve square roots differing in sign. The ones using the plus sign diverge at and are rejected.

The other one converges.

We conclude that the slanted twins are indeed congruent and that their common radius is

The following comparison between the radii of the twins and the slanted twins shows that their difference turns out to be very small.

This gives the coordinates of the centers of the slanted twins.

The following `Manipulate` shows the slanted twins and, optionally, the twins, as you vary .

In this section we generalize the shape of an arbelos by allowing the arcs to cross and by considering a 3D version. To set the context of the first of those generalizations, we need the concept of the *radical axis of two circles*.

Let be a point and be the circle . The *power* of with respect to is defined to be the real number . The power of is positive, zero, or negative depending on whether lies outside, on, or inside [12]. Let ; if the points of satisfy the equation , then an alternative way to define the power of is to evaluate . (A similar result applies if , when the circle degenerates to a line, in which case the sign of indicates whether is above, on, or below the line.)

Here is a very interesting property of the power of a point. Given a circle and a point , choose an arbitrary line through meeting the circle at points and . Then the product depends only on —it is independent of the choice of line through . This product is equal to the power of .

In the following `Manipulate`, drag the four locators to vary the size of the circle, the position of , and the slope of the line through .

Given two circles with different centers, their *radical axis* is defined to be the line consisting of all points that have equal powers with respect to each of the two circles. Proofs of the following can be found in [10].

**Theorem 3**

If two circles intersect at two points and , then their radical axis is the common secant . If two circles are tangent at , then their radical axis is their common tangent at .

**Corollary 1**

Given three circles with noncollinear centers, the three radical axes of the circles taken in pairs are distinct concurrent lines.

**Theorem 4**

The radical axis of two circles is the locus of points from which tangents drawn to both circles have the same length.

The following `Manipulate` shows two circles; one is fixed, and you can vary the center and size of the other one by dragging the locator or changing its radius with the slider. You can use the other slider to move the red point on the radical axis to illustrate theorem 4.

The following `Manipulate` illustrates two generalizations.

**Property 28**

The inscribed circles tangent to the radical axis of the side arcs and the top arc and either of the arcs of the generalized arbelos have the same radius.

Let be the length of the *gap* between the bases (so that the diameter of the top arc is ) and let be the abscissa of the intersection of the radical axis with the axis, assuming the origin is at the leftmost point of the arbelos [10].

**Theorem 5**

With the help of this theorem, we compute the value of .

We can assume without loss of generality that , , and ( can be negative). Let the inscribed circles be and . The values of these parameters are obtained as follows.

Then, although some centers can be disregarded, the radius is the same in all cases.

Finally, here are three more properties of the arbelos. See if you can guess what property is involved by experimenting with the controls [24, 25].

This first `Manipulate` lets you move the side arcs in a systematic way.

This second `Manipulate` lets you rotate a line around the point of tangency of the side arcs.

Finally, the third `Manipulate` shows an infinite family of twins.

[1] | F. van Lamoen. “Online Catalogue of Archimedean Circles.” (Jan 22, 2014) home.planet.nl/~lamoen/wiskunde/arbelos/Catalogue.htm. |

[2] | S. Garcia Diethelm. “Planar Stress Rotation” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/PlanarStressRotation. |

[3] | C. W. Dodge, T. Schoch, P. Y. Woo, and P. Yiu, “Those Ubiquitous Archimedean Circles,” Mathematical Magazine, 72(3), 1999 pp. 202-213. www.jstor.org/stable/2690883. |

[4] | H. P. Boas, “Reflection on the Arbelos,” American Mathematical Monthly, 113(3), 2006 pp. 236-249. |

[5] | H. D. Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution (D. Antin, trans.), New York: Dover Publications, 1965. |

[6] | J. Rangel-Mondragón. “Recursive Exercises II: A Paradox” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/RecursiveExercisesIIAParadox. |

[7] | R. B. Nelsen, “Proof without Words: The Area of an Arbelos,” Mathematics Magazine, 75(2), 2002 p. 144. |

[8] | A. Gadalla. “Area of the Arbelos” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/AreaOfTheArbelos. |

[9] | J. Rangel-Mondragón, “Selected Themes in Computational Non-Euclidean Geometry. Part 1. Basic Properties of Inversive Geometry,” The Mathematica Journal, 2013. www.mathematica-journal.com/2013/07/selected-themes-in-computational-non-euclidean-geometry-part-1. |

[10] | D. Pedoe, Geometry: A Comprehensive Course, New York: Dover, 1970. |

[11] | M. Schreiber. “Orthogonal Circle Inversion” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/OrthogonalCircleInversion. |

[12] | M. G. Welch, “The Arbelos,” Master’s thesis, Department of Mathematics, University of Kansas, 1949. |

[13] | L. Bankoff, “The Marvelous Arbelos,” The Lighter Side of Mathematics (R. K. Guy and R. E. Woodrow, eds.), Washington, DC: Mathematical Association of America, 1994. |

[14] | G. L. Alexanderson, “A Conversation with Leon Bankoff,” The College Mathematics Journal, 23(2),1992 pp. 98-117. |

[15] | S. Kabai. “Tangent Circle and Arbelos” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TangentCircleAndArbelos. |

[16] | G. Markowsky and C. Wolfram. “Theorem of the Owl’s Eyes” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TheoremOfTheOwlsEyes. |

[17] | P. Y. Woo, “Simple Constructions of the Incircle of an Arbelos,” Forum Geometricorum, 1, 2001 pp. 133-136. forumgeom.fau.edu/FG2001volume1/FG200119.pdf. |

[18] | B. Alpert. “Archimedes’ Twin Circles in an Arbelos” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ArchimedesTwinCirclesInAnArbelos. |

[19] | J. Rangel-Mondragón. “Twins of Arbelos and Circles of a Triangle” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TwinsOfArbelosAndCirclesOfATriangle. |

[20] | H. Okumura, “More on Twin Circles of the Skewed Arbelos,” Forum Geometricorum, 11, 2011 pp. 139-144. forumgeom.fau.edu/FG2011volume11/FG201114.pdf. |

[21] | E. W. Weisstein. “Point-Line Distance—2-Dimensional” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/Point-LineDistance2-Dimensional.html. |

[22] | L. Bankoff, “Are the Twin Circles of Archimedes Really Twins?,” Mathematics Magazine, 47(4), 1974 pp. 214-218. |

[23] | F. Power, “Some More Archimedean Circles in the Arbelos,” Forum Geometricorum, 5, 2005 pp. 133-134. forumgeom.fau.edu/FG2005volume5/FG200517.pdf. |

[24] | A. V. Akopyan, Geometry in Figures, CreateSpace Independent Publishing Platform, 2011. |

[25] | H. Okumura and M. Watanabe, “Characterizations of an Infinite Set of Archimedean Circles,” Forum Geometricorum, 7, 2007 pp. 121-123. forumgeom.fau.edu/FG2007volume7/FG200716.pdf. |

J. Rangel-Mondragón, “The Arbelos,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-5. |

Jaime Rangel-Mondragón received M.Sc. and Ph.D. degrees in applied mathematics and computation from the University College of North Wales in Bangor, UK. He has been a visiting scholar at Wolfram Research, Inc. and has held positions in the Faculty of Informatics at UCNW, the College of Mexico, the Center for Research and Advanced Studies, the Monterrey Institute of Technology, the Queretaro Institute of Technology, and the University of Queretaro in Mexico, where he is presently a member of the Faculty of Informatics. His current research includes combinatorics, the theory of computing, computational geometry, urban traffic, and recreational mathematics.

**Jaime Rangel-Mondragón**

*UAQ, Facultad de Informatica
Queretaro, Qro. Mexico*

Generally, to carry out a regression procedure one needs to have a model , an error definition , and the probability density function of the error . Considering the set as measurement points, the maximum likelihood approach aims at finding the parameter vector that maximizes the likelihood of the joint error distribution. Assuming that the measurement errors are independent, we should maximize (see eg. [1])

(1) |

Instead of maximizing this objective, we minimize

(2) |

Consider the Gaussian-type error distribution as ; then our estimator is

(3) |

In our case the model is a line,

(4) |

It can be seen that (in the case of Gaussian-type measurement noise) only the type of the error model determines the parameter values, since we should always minimize the least squares of the errors. There are different error models, which can be applied to fitting a line in a least-squares sense. The error model frequently employed, assuming an error-free independent variable , is the ordinary least squares model ()

(5) |

Similarly, one may also consider an error-free dependent variable . Then the error model () is

(6) |

These approaches are called the *algebraic approach*.

Another error model considers the geometrical distance between the data point and the line to be fitted. This type of fitting is also known as *orthogonal regression*, since the distances of the sample points from the line are evaluated by computing the orthogonal projection of the measurements on the line itself. The error in this case [2] is

(7) |

This *geometrical approach* or *total least squares* () approach can also be considered as an optimization problem with constraints; namely, one should minimize the errors in both variables [3]:

(8) |

under the conditions

(9) |

In addition, one can also combine and to construct an error model. The first possibility is to consider the geometric mean of these two types of errors,

(10) |

These error models are illustrated in Figure 1.

**Figure 1.** The different error models in the case of fitting a straight line.

This model is also called the *least* *geometric mean deviation* approach or model (see [4)]. As a second possibility, one may consider and as competing functions of the parameters and find their Pareto-front representing a set of optimal solutions for the parameters . Since this multi-objective problem is convex, the objective can be expressed as a linear combination of these error functions, namely

(11) |

where is a parameter, , and the set of optimal solutions of the parameters belonging to the Pareto-front is . You can choose the value of depending on your trade-off preference between and [5].

Symbolic computation can be used to avoid direct minimization and to get an explicit formula for the estimated parameters. We apply the *Mathematica* function `SuperLog` developed in [6], which uses pattern matching that enhances *Mathematica*’s ability to simplify expressions involving the natural logarithm of a product of algebraic terms.

Let us activate this function.

Then this is the ML estimator for Gaussian-type noise.

Now let us consider the problem.

Here are the necessary conditions for the optimum.

Let us introduce the following constants:

(12) |

(13) |

(14) |

(15) |

(16) |

In those terms, here are the necessary conditions for the optimum.

Then this is the optimal solution of the parameters.

Although the equation system for the parameters of is linear, for other error models we get a multivariable algebraic system. Now consider the problem. Here is the maximum likelihood function.

Therefore here is the equation system to be solved.

Since , the conditions are as follows.

A Gröbner basis solves this system, eliminating .

Since the second equation is linear, it is reasonable to compute first, then .

The error model also leads to a second-order polynomial equation system. Now here is the ML estimator.

Consequently, here is the system to be solved for the parameters.

Assume .

Again a Gröbner basis gives a second-order system.

When is known, the other parameter can be computed.

In the case of the Pareto approach, the system is already fourth order.

Here is the system.

Here is the system in compact form.

Here is the Gröbner basis for the first parameter.

Assume that .

After solving this polynomial for , the other parameter can be solved from the second equation, which is linear in .

Consider some data on rainfall (in mm) and the resulting groundwater level changes (in cm) from a landslide along the Ohio River Valley near Cincinnati, Ohio [7].

There are 14 measurements.

This displays the measured data.

**Figure 2.** The measured data: rainfall versus water level change in dimensional form.

The constants , , , , and in equations (12) to (16) are needed.

This separates the data.

This transforms the data into dimensionless form.

**Figure 3.** The measured data: rainfall versus water level change in dimensionless form.

Now the constants can be computed.

Here are the estimated parameters employing the explicit solutions.

This checks the result.

Figure 4 shows the estimated line with the sample points.

**Figure 4.** The sample points with the line estimated with .

Here are the first and second parameters.

Here is a check of this result on the basis of the definition. Equation (8) gives the objective function.

The constraints are .

The unknown variables are not only the parameters, but the adjustments as well.

This uses a built-in global optimization method. (This takes a long time to compute.)

The estimation gives a result quite different from the model; see Figure 5.

**Figure 5.** The lines estimated with the (red) and (green) models.

Since the constraints are linear, the optimization can be written in unconstrained form, reducing the original number of variables to .

Now here is the first parameter.

This uses the result.

Here is a numerical check of the objective.

Figure 6 shows this result together with the and models.

**Figure 6.** The lines estimated with the (red), (green), and (blue) models.

The first parameter is a fourth-order polynomial.

The best trade-off between and is to let .

This is the real positive solution.

Using this value gives the second parameter.

We compute the solution using direct global minimization. Here is the objective.

This gives the result.

Figure 7 shows this solution with the results of the other models.

**Figure 7.** The lines estimated with the (red), (green), and (blue) models, and the Pareto approach with (magenta).

The numerical computations show that the formulas developed by an ML estimator via symbolic computation to determine the parameters of a straight line to be fitted provide correct results and require considerably less computation time than the direct methods based on global minimization of the residuals. Our examples also illustrate that the , , and Pareto approaches give more realistic solutions than the traditional , since Figure 7 shows there are at least two outliers in the sample set.

[1] | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed., Cambridge: Cambridge University Press, 1992. |

[2] | M. Zuliani. “RANSAC for Dummies.” (Jan 10, 2014) vision.ece.ucsb.edu/~zuliani/Research/RANSAC/docs/RANSAC4Dummies.pdf. |

[3] | B. Schaffrin, “A Note on Constrained Total Least-Squares Estimation,” Linear Algebra and Its Applications, 417(1), 2006 pp. 245-258. doi:10.1016/j.laa.2006.03.044. |

[4] | C. Tofallis, “Model Fitting for Multiple Variables by Minimising the Geometric Mean Deviation,” in Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications (S. Van Huffel and P. Lemmerling, eds.), Dordrecht: Kluwer, 2002. |

[5] | B. Paláncz and J. L. Awange, “Application of Pareto Optimality to Linear Models with Errors-in-All-Variables,” Journal of Geodesy, 86(7), 2012 pp. 531-545.doi:10.1007/s00190-011-0536-1. |

[6] | C. Rose and M. D. Smith, “Symbolic Maximum Likelihood Estimation with Mathematica,” The Statistician, 49(2), 2000 pp. 229-240. www.jstor.org/stable/2680972. |

[7] | W. C. Haneberg, Computational Geosciences with Mathematica, Berlin: Springer, 2004. |

B. Paláncz, “Fitting Data with Different Error Models,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-4. |

Béla Paláncz received his D.Sc. degree in 1993 from the Hungarian Academy of Sciences and has wide-ranging experience in teaching and research (RWTH Aachen, Imperial College London, DLR Köln, and Wolfram Research). His main research fields are mathematical modeling and symbolic-numeric computation.

**Béla Paláncz**

*Department of Photogrammetry and Geoinformatics,
Budapest University of Technology and Economics
1521 Budapest, Hungary *

The Karush-Kuhn-Tucker equations (under suitable assumptions) provide necessary and sufficient conditions for the solution of the problem of maximizing (minimizing) a concave (convex) function.

For an excellent reference, see the tutorial in [2]. Here we modify the code of [1] by correcting minor typos, simplifying, and letting the user specify restrictions on the exogenous parameters of the model.

The inputs of `KT` are the objective function to be maximized, the list of constraints, and the list of choice variables. Here is an example from consumer choice theory: maximize a utility function, subject to a budget constraint.

Several of the solutions do not make economic sense, because they do not use the fact that the income, price of good , and price of good are all positive. However, `KT` lets the user specify restrictions on the exogenous parameters of the model.

An important advantage of `KT` over other optimization functions (such as `Maximize` or `Minimize`) is that `KT` returns the value of the Kuhn-Tucker multipliers. These multipliers have an important economic interpretation: they are shadow prices for the constrained resources. In the above example, for instance, the value of is the “infinitesimal” increment in the utility function of the consumer that is generated when the budget constraint is relaxed by increasing the consumer’s income by an “infinitesimal amount.”

Nash equilibrium is the main solution concept in game theory. It is a crucial tool for economics and political science models. Essentially, a Nash equilibrium is a profile of strategies (one strategy for each player), such that if a player takes the choices of the others as given (i.e. as parameters), then the player’s strategy must maximize his or her payoff.

The function `Nash` takes as input the payoff function of player 1, the payoff function of player 2, and the actions available to players 1 and 2. It returns the entire set of Nash equilibria.

There are many versions of Colonel Blotto’s game; this is a simple one taken from [3]. General A (row player) has three divisions to defend a city; she has to choose how many divisions to place at the north road and how many divisions at the south road. General B (column player) has two divisions to try to invade the city; he also has to choose how many divisions to be assigned to the north road and how many to the south road. If General A has at least as many divisions as General B at a given road, General A wins the battle there (defense is favored in the case of a tie). To win the game, however, A must defeat B on both battlefields. Thus, A has four possible strategies and B has three strategies. The table below summarizes the players’ strategies and payoffs (, for the whole campaign). For example, in the first row and first column the entry is , which means A won and B lost; A chose three divisions for the north road and none for the south road; B chose two for the north and none for the south. Because and , A won both battles.

A Nash equilibrium for this game is a probability distribution over strategies; use `P` for the probabilities chosen by General A and `Q` for the probabilities chosen by General B.

The game has many Nash equilibria, but we still can make predictions: General B is never going to spread his forces evenly (the probability of his second strategy is zero in any equilibrium, ); with probability , B’s two divisions are placed at the north road () and with probability , they are placed at the south road (). As for General A, the probability that she places all of her three divisions on one front is less than half (i.e. and ). Also, the probability that General A places two or more divisions at the north (or south) is always equal to half (i.e. and ).

This game is also borrowed from [3]. A deck has two cards, one high and one low. Each player places one dollar into the pot. Player 1 gets one card from the deck. Player 2 does not see Player 1’s card. Player 1 decides whether to raise (by placing another dollar in the pot) or not raise. Player 2 observes 1’s action and then has to decide whether to match the bet or fold. If Player 2 folds, then Player 1 wins the contents of the pot. However, if Player 2 matches, Player 2 places another dollar into the pot if Player 1 had previously raised. Player 1 reveals her card. If it is the high card, Player 1 wins the pot; otherwise, Player 2 wins it.

See Figure 1 for the corresponding game tree. We introduce a fictitious player, Nature, who randomly decides if the card is high or low. We depict the bimatrix representation of the game. Player 1 has four strategies: always raise (RR), always not raise (NN), raise if the card is high and not otherwise (RN), and not raise if the card is high and raise otherwise (NR). Player 2 also has four strategies: always match (MM), always fold (FF), match only if Player 1 raised (MF), and fold only if Player 1 raised (FM). For simplicity, in the bimatrix representation, we write the expected payoffs of Player 1 and omit Player 2’s payoffs (this is without loss of generality in zero-sum games).

**Figure 1.** Game tree of the card game.

In this case, the Nash equilibrium delivers a sharp prediction. When Player 1 has the high card, she always raises (), but when she has the low card, she bluffs with probability (the probability of RR is ). When Player 1 does not raise, Player 2 always matches (). If Player 1 raises, Player 2 still may match, but with probability (the probability of always matching MM is ).

We extended the code of [1] to solve for Kuhn-Tucker conditions with additional assumptions on parameters and, more importantly, using the Kuhn-Tucker equations we provide a program to compute all the Nash equilibria of finite bimatrix games.

We presented a program to compute the set of all Nash equilibria in finite bimatrix games. Its intended goal is as a classroom tool for students and instructors. Needless to say, the code is not efficient. For larger inputs (say bimatrix games with five or more actions per player), `Reduce` often fails to solve the system of Kuhn-Tucker equations. For optimizing algorithms, we suggest [4]. Nevertheless, with continuous improvement of hardware and algorithms for solving semialgebraic systems (see [5]), these methods may become useful for research applications sooner than we think. Finally, as algorithmic game theory courses become more popular in computer science departments, it seems that the time to bring computational methods and algorithms to economics departments is already overdue.

[1] | F. J. Kampas, “Tricks of Using Reduce to Solve Khun-Tucker Equations,” The Mathematica Journal, 9(4), 2005 pp. 686-689.www.mathematica-journal.com/issue/v9i4/contents/Tricks9-4/Tricks9-4_ 2.html. |

[2] | M. J. Osborne. “Optimization: The Kuhn-Tucker Conditions for Problems with Inequality Constraints,” from Mathematical Methods for Economic Theory: A Tutorial. (Jan 8, 2014)www.economics.utoronto.ca/osborne/MathTutorial/MOIF.HTM. |

[3] | M. Osborne, “An Introduction to Game Theory,” New York: Oxford University Press, 2004. |

[4] | R. D. McKelvey, A. M. McLennan, and T. L. Turocy. “Gambit: Software Tools for Game Theory.” (Jan 8, 2014) www.gambit-project.org. |

[5] | Wolfram Research, “Real Polynomial Systems” from Wolfram Mathematica Documentation Center—A Wolfram Web Resource.reference.wolfram.com/mathematica/tutorial/RealPolynomialSystems.html. |

S. O. Parreiras, “Using Reduce to Compute Nash Equilibria,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-3. |

Sérgio O. Parreiras is an associate professor at the UNC-Chapel Hill Department of Economics. His research focus is on game theory and its applications to auctions, mechanism design, and contests. He is also interested in computational economics, general equilibrium theory, algorithmic game theory, and evolutionary anthropology.

**Sérgio O. Parreiras**

*UNC, Department of Economics
Gardner Hall, 200B
Chapel Hill, N.C. 27599-3305
*

The infinite Fibonacci word,

is certainly one of the most studied words in the field of combinatorics on words [1-4]. It is the archetype of a Sturmian word [5]. The word can be associated with a fractal curve with combinatorial properties [6-7].

This article implements *Mathematica* programs to generate curves from and a set of drawing rules. These rules are similar to those used in L-systems.

The outline of this article is as follows. Section 2 recalls some definitions and ideas of combinatorics on words. Section 3 introduces the Fibonacci word, its fractal curve, and a family of words whose limit is the Fibonacci word fractal. Finally, Section 4 generalizes the Fibonacci word and its Fibonacci word fractal.

The terminology and notation are mainly those of [5] and [8]. Let be a finite alphabet, whose elements are called symbols. A word over is a finite sequence of symbols from . The set of all words over , that is, the free monoid generated by , is denoted by . The identity element of is called the empty word. For any word , denotes its length, that is, the number of symbols occurring in . The length of is taken to be zero. If and , then denotes the number of occurrences of in .

For two words and in , denote by the concatenation of the two words, that is, . If , then ; moreover, by denote the word ( times). A word is a subword (or factor) of if there exist such that . If , then and is called a prefix of ; if , then and is called a suffix of .

The reversal of a word is the word and . A word is a palindrome if .

An infinite word over is a map , written as . The set of all infinite words over is denoted by .

**Example 1**

The word , where if is a prime number and otherwise, is an example of an infinite word. The word is called the characteristic sequence of the prime numbers. Here are the first 50 terms of .

**Definition 1**

There is a special class of words with many remarkable properties, the so-called Sturmian words. These words admit several equivalent definitions (see, e.g. [5], [8]).

**Definition 2**

Let . Let , the complexity function of , be the map that counts, for all integer , the number of subwords of length in . An infinite word is a Sturmian word if for all integer .

For example, .

Since for any Sturmian word, , Sturmian words have to be over two symbols. The word in example 1 is not a Sturmian word because .

Given two real numbers , with irrational and , , define the infinite word as . The numbers and are the slope and the intercept, respectively. This word is called mechanical. The mechanical words are equivalent to Sturmian words [5]. As a special case, gives the characteristic words.

**Definition 3**

On the other hand, note that every irrational has a unique continued fraction expansion

where each is a positive integer. Let be an irrational number with and for . To the directive sequence , associate a sequence of words defined by , , , .

Such a sequence of words is called a standard sequence. This sequence is related to characteristic words in the following way. Observe that, for any , is a prefix of , which gives meaning to as an infinite word. In fact, one can prove that each is a prefix of for all and [5].

**Definition 4**

Fibonacci words are words over defined inductively as follows: , , and , for . The words are referred to as the finite Fibonacci words. The limit

(1) |

It is clear that , where is the Fibonacci number, recalling that the Fibonacci number is defined by the recurrence relation for all integer and with initial values . The infinite Fibonacci word is a Sturmian word [5]; exactly, , where is the golden ratio.

Here are the first 50 terms of .

**Definition 5**

The Fibonacci word satisfies and for all .

Here are the first nine finite Fibonacci words.

**Definition 6**

The following proposition summarizes some basic properties about the Fibonacci word.

**Proposition 1**

- The words 11 and 000 are not subwords of the Fibonacci word.
- Let be the last two symbols of . For , if is even and if is odd.
- The concatenation of two successive Fibonacci words is almost commutative; that is, and have a common prefix of length , for all .
- is a palindrome for all .
- For all , , where ; that is, exchanges the two last symbols of .

The Fibonacci word can be associated with a curve using a drawing rule. A particular action follows on the symbol read (this is the same idea as that used in L-systems [9]). In this case, the drawing rule is called “*the odd-even drawing rule*” [7].

**Definition 7**

The Fibonacci curve, denoted by , is the result of applying the odd-even drawing rule to the word . The Fibonacci word fractal is defined as

The program `LShow` is adapted from [10] to generate L-systems.

Figure 1 shows an L-system interpretation of the odd-even drawing rule.

**Figure 1. **Interpretation of the odd-even drawing rule.

Here are the curves for .

The next proposition about properties of the curves and comes directly from the properties of the Fibonacci word from Proposition 1. More properties can be found in [7].

**Proposition 2**

- is composed only of segments of length 1 or 2.
- The number of turns in the curve is the Fibonacci number .
- The curve is similar to the curve .
- The curve is symmetric.
- The curve is composed of five curves: , where is the result of applying the odd-even drawing rule to the word .

The next figure shows the curve and the five curves; here .

The Fibonacci word and other words can be derived from the dense Fibonacci word, which was introduced in [7].

**Definition 8**

(2) |

Given a drawing rule, the global angle is the sum of the successive angles generated by the word through the rule. With the natural drawing rule, , , , then .

For a drawing rule, the resulting angle of a word is the function that gives the global angle. A morphism preserves the resulting angle if for any word , ; moreover, a morphism inverts the resulting angle if for any word , .

The dense Fibonacci word is strongly linked to the Fibonacci word fractal because can generate a whole family of curves whose limit is the Fibonacci word fractal [7]. All that is needed is to apply a morphism to that preserves or inverts the resulting angle.

Here are some examples.

Here are some examples with other angles.

This section introduces a generalization of the Fibonacci word and the Fibonacci word fractal [11].

**Definition 9**

The 2-Fibonacci word is the classical Fibonacci word. Here are the first six -Fibonacci words.

The following proposition relates the Fibonacci word to .

**Proposition 3**

(3) |

**Definition 10**

The -Fibonacci numbers are the Fibonacci numbers and the -Fibonacci numbers are the Fibonacci numbers shifted by one. The following table shows the first terms in the sequences and their reference numbers in the On-Line Encyclopedia of Sequences (OIES) [12].

**Proposition 4**

- The word 11 is not a subword of the -Fibonacci word, .
- Let be the last two symbols of . For , if is even and if is odd, .
- The concatenation of two successive -Fibonacci words is almost commutative; that is, and have a common prefix of length for all and .
- is a palindrome for all .
- For all , , where .

**Theorem 1**

For the proof, see [11]. This theorem implies that -Fibonacci words are Sturmian words.

Note that

where is the golden ratio.

**Definition 11**

The Fibonacci curve, denoted by , is the result of applying the odd-even drawing rule to the word . The -Fibonacci word fractal is defined as

Here are the curves for .

**Proposition 5**

- The Fibonacci fractal is composed only of segments of length 1 or 2.
- The curve is similar to the curve .
- The curve is composed of five curves: .
- The curve is symmetric.
- The scale factor between and is .

This section applies the above ideas to generate new curves from characteristic words (see

Definition 3).

**Conjecture 1**

Here are seven examples.

The first author was partially supported by Universidad Sergio Arboleda under grant number USA-II-2012-14. The authors would like to thank Borut Jurčič-Zlobec from Ljubljana University for his help during the development of this article.

[1] | J. Cassaigne, “On Extremal Properties of the Fibonacci Word,” RAIRO—Theoretical Informatics and Applications, 42(4), 2008 pp. 701-715. doi:10.1051/ita:2008003. |

[2] | W. Chuan, “Fibonacci Words”, Fibonacci Quarterly, 30(1), 1992 pp. 68-76. www.fq.math.ca/Scanned/30-1/chuan.pdf. |

[3] | W. Chuan, “Generating Fibonacci Words,” Fibonacci Quarterly, 33(2), 1995 pp. 104-112. www.fq.math.ca/Scanned/33-2/chuan1.pdf. |

[4] | F. Mignosi and G. Pirillo, “Repetitions in the Fibonacci Infinite Word,” RAIRO—Theoretical Informatics and Applications, 26(3), 1992 pp. 199-204. |

[5] | M. Lothaire, Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications), Cambridge: Cambridge University Press, 2005. |

[6] | A. Blondin Massé, S. Brlek, A. Garon, and S. Labbé, “Two Infinite Families of Polyominoes That Tile the Plane by Translation in Two Distinct Ways,” Theoretical Computer Science, 412(36), 2011 pp. 4778-4786. doi:10.1016/j.tcs.2010.12.034. |

[7] | A. Monnerot-Dumaine, “The Fibonacci Word Fractal,” preprint, 2009. hal.archives-ouvertes.fr/hal-00367972/fr. |

[8] | J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge: Cambridge University Press, 2003. |

[9] | P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, New York: Springer-Verlag, 1990. |

[10] | E. Weisstein. “Lindenmayer System” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/LindenmayerSystem.html. |

[11] | J. Ramírez, G. Rubiano, and R. de Castro, “A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake,” 2013. arxiv:1212.1368v2. |

[12] | OEIS Foundation, Inc. “The On-Line Encyclopedia of Integer Sequences.” (Aug 9, 2013) oeis.org. |

J. L. Ramírez and G. N. Rubiano, “Properties and Generalizations of the Fibonacci Word Fractal,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-2. |

**José L. Ramírez**

*Instituto de Matemáticas y sus Aplicaciones
Universidad Sergio Arboleda
Calle 74 no. 14 – 14 Bogotá, Colombia*

**Gustavo N. Rubiano **

*Departamento de Matemáticas
Universidad Nacional de Colombia
AA 14490, Bogotá, Colombia*

Bayesian statistics are an orderly way of finding the likelihood of a model from data, using the likelihood of the data given the model. From spam detection to medical diagnosis, spelling correction to forecasting economic and demographic trends, Bayesian statistics have found many applications, and even praise as mental heuristics to avoid overconfidence. However, at first glance Bayesian statistics suffer from an apparent limit: they can only make inferences about known factors, bounded to conditions seen within the data, and have nothing to say about the likelihood of new phenomena [3]. In short, Bayesian statistics are apparently withheld to inferences about the parameters of the model they are provided.

Instead of taking priors over factors of the model itself, we can say that we are taking priors over factors in the process involving how the data was generated. These stochastic process priors give the modeler a way to talk about factors that have not been directly observed. These nonobservable factors include the likely rate at which further factors might be seen, given further observation and underlying categories or structures that might generate the data being observed. For example, in statistics problems we are often presented with drawing marbles of different colors from a bag, and given randomly drawn samples, we might talk about the most likely composition of the bag and the range of likely compositions. However, suppose we had a number of bags, and we drew two marbles each from three of them, discovering two red marbles, two green marbles, and two yellow marbles [4]. If we were to draw marbles from yet another bag, we might expect two marbles identical in color, of a color we have not previously observed. We do not know what this color is, and in this sense we have made a nonparametric inference about the process that arranged the marbles between bags.

The ability to talk about nonobserved parameters is a leap in expressiveness, as instead of explicitly specifying a model for all parameters, a model utilizing infinite processes expands to fit the given data. This should be regarded similarly to the advantages afforded by linked data structures in representing ordinary data. A linked list has a potentially infinite capacity; its advantage is not that we have an infinite memory, but an abstract flexibility to not worry too much about maintaining its size appropriately. Similarly, an infinite prior models the growth we expect to discover [5].

Here are two specific processes that are useful for a number of different problems. These two processes are good for modeling unknown discrete categories and sets of features, respectively. In both of these processes, suppose that we can take samples so that there are no dependencies in the order that we took them, or in other words that the samples are exchangeable. Both of these processes also make use of a concentration parameter, . As we look at more samples, we expect the number of new elements we discover to diminish, but not disappear, as our observations establish a lower frequency of occurrence for unobserved elements. The concentration parameter establishes the degree to which the proportions are concentrated, with low indicating a distribution concentrated on a few elements, and high indicating a more dispersed concentration.

First, let us look into learning an underlying system of categories. In a fixed set of categories of particular likelihood, the probability of a given sample in a particular category corresponds to the multinomial distribution, the multiparameter extension of the Bernoulli distribution. The conjugate prior, or the distribution that gives a Bayesian estimate of which multinomial distribution produced a given sample, is the Dirichlet distribution, itself the multivariable extension of the beta distribution. To create an infinite Dirichlet distribution, or rather a Dirichlet process, one can simply have a recursive form of the beta where the likelihood of a given category is . To use a Dirichlet process as a prior, it is easier to manipulate in the form of a Chinese restaurant process (CRP) [6]. Suppose we want to know the likelihood that the sample is a member of category . If the category is new, then that probability corresponds to the size of the concentration parameter in ratio to the count of the samples taken:

The implementation of this function is straightforward. The use of a parameterized random number function allows for the use of the algorithm in common random number comparison between simulation scenarios [7], as well as for estimation through Markov chain Monte Carlo, about which more will be said later.

In the second process, suppose we are interested in the sets of features observed in sets of examples. For example, suppose we go to an Indian food buffet and are unfamiliar with the dishes, so we observe the selected items that our fellow patrons have chosen. Supposing one overall taste preference, we might say that the likelihood of a dish’s being worth selecting is proportional to the number of times it was observed, but if there are not many examples we should also try some additional dishes that were not tried previously. This process, called the Indian buffet process [8], turns out to be equivalent to a beta process prior [9]. Suppose we want to know the likelihood of whether a given feature is going to be found in the sample. Then, the likelihoods can be calculated directly from other well-understood distributions:

Both of these processes are suitable as components in mixture models. Suppose we are conducting a phone poll of a city and ask the citizens we talk to about their concerns. Each person will report their various civic travails. We expect for each person to have their own varying issues, but also for there to be particular groups of concern for different neighborhoods and professional groups. In other words, we expect to see an unknown set of features emerge from an unknown set of categories. Then, we might use a CRPIBP mixture distribution to help learn those categories from the discovered feature sets.

Nonparametric inference tasks are particularly suited for computational support. What we would like to do is describe a space of potential mixture models that may describe the underlying data-generation processes and allow the inference of their likelihood without explicitly generating the potential structures of that space. Probabilistic programming is the use of language-specific support to aid in the process of statistical inference. This article shows that *Mathematica* has features that readily enable the sort of probabilistic programming that supports nonparametric inference.

Probabilistic programming is the use of language-specific support to aid in the process of statistical inference. Unlike statistical libraries, the structure of the programming language itself is used in the inference process [10]. Although *Mathematica* increasingly has the kinds of structures that support probabilistic programming, we are not going to focus on those features here. Instead, we will see how *Mathematica*’s natural capacity for memoization allows it to be very easily extended to write probabilistic programs that use stochastic memoization as a key abstraction. In particular, we are going to look at Church, a Lisp-variant with probabilistic query and stochastic memoization constructs [11]. Let us now explain stochastic memoization and then look at how to implement Metropolis-Hastings querying, which uses memoization to help implement Markov chain Monte Carlo-driven inference.

Stochastic memoization simply means remembering probabilistic events that have already occurred. Suppose we say that is the first flip of coin `c`. In the first call, it may return `Heads` or `Tails`, depending on a likelihood imposed to coin `c`, but in either case it is constrained in later calls to return the same value. Once undertaken, the value of a particular random event is determined.

In Church, this memoization is undertaken explicitly through its `mem` operator. Church’s flip function designates a Bernoulli trial with the given odds, with return values 0 and 1. Here is an example of a memoized fair coin flip in Church.

(define coinflip (mem (lambda (coin flip) (flip 0.5))))

*Mathematica* allows for a similar memoization by incorporating a `Set` within a `SetDelayed`.

Let us now look to a more complicated case. Earlier, we discussed the Dirichlet process. Church supports a DPmem operator for creating functions that when given a new example either returns a previously obtained sample according to the CRP or takes a new sample, depending upon the category assignment, and returns the previously seen argument. Here is a similar function in *Mathematica*, called `GenerateMemCRP`. Given a random function, we first create a memoized version of that function based on the category index of the CRP. Then, we create an empty initial CRP result, for which a new sample is created and memoized every time a new input is provided, potentially also resampling the provided function if a prediscovered category is provided.

For example, let us now take a sampling from categories that have a parameter distributed according to the standard normal distribution. Here we see outputs in a typical range for a standard normal, but with counts favoring resampling particular results according to the sampled frequency of the corresponding category.

Memoization implies that if we provide the same inputs, we get the same results.

Inference is the central operation of probabilistic programming. Conditional inference is implemented in Church through its various query operations. These queries uniformly take four sets of arguments: query algorithm-specific parameters, a description of the inference problem, a condition to be satisfied, and the expression we want to know the distribution of given that condition. Let us motivate the need for a *Mathematica* equivalent to the Church query operator by explaining other queries that are trivial to implement in *Mathematica* but that are not up to certain inference tasks.

Direct calculation is the most straightforward approach to conditional inference. However, sometimes we cannot directly compute the conditional likelihood, but instead have to sample the space. The easiest way to do so is rejection sampling, in which we generate a random sample for all random parameters to see if it meets the condition to be satisfied. If it does, its value is worth keeping as a sample of the distribution, and if it does not, we discard it entirely, proceeding until we are satisfied that we have found the distribution we intend.

There is a problem with rejection sampling, namely that much of the potential model space might be highly unlikely and that we are throwing away most of the samples. Instead of doing that, we can start at a random place but then, at each step, use that sample to find a good sample for the underlying distribution [12]. So, for a sample , we are interested in constructing a transition operator yielding a new sample , and constructing that operator such that for the underlying distribution , the transition operator is invariant with respect to distribution , or in other words, that the transition operator forms a Markov chain. For our transition operator, we first choose to generate a random proposal, , where a simple choice is the normally distributed variation along all parameters , and then accept that proposal with likelihood , so that we are incorporating less-likely samples at exactly the rate the underlying distribution would provide. After some initial samples of random value, we will have found the region for which the invariance property holds. Due to the use of applying random numbers to a Markov chain, this algorithm is called Markov chain Monte Carlo, or MCMC.

The following procedure is intended to be the simplest possible implementation of MCMC using memoization (for further considerations see [13, 14]). There is a trade-off in the selection of , such that if it is too large, we rarely accept anything and would effectively be undertaking rejection sampling, but if it is too small, we tend to stay in a very local area of the algorithm. One way to manage this trade-off is to control by aiming for a given rejection rate, which is undertaken here.

We now see why we constructed the CRP functions to accept random number functions: it lets us create evaluation functions suitable for MCMC.

Let us look to see how we might apply these examples. First, we are going to look at the infinite relational model, which demonstrates how to use the CRP to learn underlying categories from relations. Then, we will look at learning arithmetic expressions based upon particular inputs and outputs, which demonstrates using probabilistic programming in a recursive setting.

Suppose we are given some set of relations in the form of predicates, and we want to infer category memberships based on those relations. The infinite relational model (IRM) can construct infinite category models for processing arbitrary systems of relational data [15]. Suppose now we have some specific instances of objects, , and a few specific statements about whether a given -ary relation, , holds between them or not. Given a prior of how concentrated categories are, , and a prior for the sharpness of a relation to holding between members of various types, , we would like to learn categories and relational likelihoods , such that we can infer category memberships for objects and the likelihood of unobserved relationships holding between them, which corresponds to the following model structure.

Below we provide a sampler to calculate this, which first sets up memoization for category assignment, next memoization for relational likelihood, and then a function for first evaluating the object to category sampling and then the predicate/category sampling. Then, the function merely calculates the likelihood sampling along the way, returning the object to category memberships and the likelihood.

To understand how this works, let us look at an example. Suppose that there are two elementary schools, one for girls and one for boys, and that they both join together for high school. However, there is a new teacher who does not know this about the composition of incoming classes. This teacher finds another teacher and asks if they know who knows whom. This more experienced teacher says yes, but not why, and the younger teacher asks a series of questions about who knows whom, to the confusion of the older teacher, who does not understand why the younger teacher does not know (we have all had conversations like this). One potential set of such questions might yield the following answers. Notice that there is a deficiency in these questions; namely, the new teacher never asks if a boy knows a girl.

Given these samples, let us now perform a Metropolis-Hastings query to see if we can recover these categories. The result of a particular sample is a list of rules, where the left side of each rule is the likelihood of the given predicates to hold given the sampled model, and the right side is the sampled model in a two-element list. In this two-element list characterizing the sampled model, the first element is the result as provided by the evaluation method, and the second element contains the random values parameterizing the model.

Given these samples, let us now find a list of categories that fit them. Normalize the weight of each example by its likelihood, filter out the sampling information, and gather the common results together.

Removing the specific category assignments and determining for each person whether they are in the same category as each other, we see that we have a complete and accurate estimate for who knows whom.

There is no more idiomatic example of probabilistic programming than probabilistically generated programs. Here, we show how to implement learning simple arithmetic expressions. A Church program for undertaking this is as follows [16]. First, it defines a function for equality that is slightly noisy, creating a gradient that is easier to learn than strict satisfaction. Next, it creates a random arithmetic expression of nested addition and subtraction with a single variable and integers from 0 to 10 as terminals. Then, it provides a utility for evaluating symbolically constructed expressions. Finally, it demonstrates sampling a program with two results that are consistent with adding 2 to the input.

Let us now construct an equivalent

Next, here is an equivalent program for generating random programs. By recursively indexing each potential branch of the program, we can assure that common random number and MCMC algorithms will correctly assign a random number corresponding to that exact part of the potential program. We also explicitly limit the size of the tree.

Now we make a Metropolis-Hastings query and process the results down to the found expression and its calculated likelihood.

Given only one example, we cannot tell very much, but are pleased that the simplest, yet correct, function is the one rated most likely. Interestingly, the first six expressions are valid despite the noisy success condition.

With two inputs, the only viable expression is the one found.

We have now seen how to implement nonparametric Bayesian inference with *Mathematica*’s memoization features. Nonparametric Bayesian inference extends Bayesian inference to processes, allowing for the consideration of factors that are not directly observable, creating flexible mixture models with similar advantages to flexible data structures. We see that *Mathematica*’s capacity for memoization allows for the implementation of nonparametric sample generation and for Markov chain sampling. This capacity was then demonstrated with two examples, one for discovering the categories underlying particular observed relations and the other for generating functions that matched given results.

Probabilistic programming is a great way to undertake nonparametric Bayesian inference, but one should not confuse language-specific constructs with the language features that allow one to undertake it profitably. Through *Mathematica*’s memoization capabilities, it is readily possible to make inferences over flexible probabilistic models.

[1] | DARPA. “Probabilistic Programming for Advancing Machine Learning (PPAML).” Solicitation Number: DARPA-BAA-13-31. (Aug 8, 2013) www.fbo.gov/utils/view?id=a7bdf07d124ac2b1dda079de6de2eb78. |

[2] | B. Cronin. “What Is Probabilistic Programming?” O’Reilly Radar (blog). (Aug 8, 2013) radar.oreilly.com/2013/04/probabilistic-programming.html. |

[3] | D. Fidler, “Foresight Defined as a Component of Strategic Management,” Futures, 43(5), 2011 pp. 540-544. doi:10.1016/j.futures.2011.02.005. |

[4] | C. Kemp, A. Perfors, and J. B. Tenenbaum, “Learning Overhypotheses with Hierarchical Bayesian Models,” Developmental Science, 10(3), 2007 pp. 307-321.doi:10.1111/j.1467-7687.2007.00585.x. |

[5] | M. I. Jordan, “Bayesian Nonparametric Learning: Expressive Priors for Intelligent Systems,” in Heuristics, Probability, and Causality: A Tribute to Judea Pearl, (R. Dechter, H. Geffner, and J. Y. Halpern, eds.) London: College Publications, 2010. |

[6] | J. Pitman, Combinatorial Stochastic Processes (Lecture Notes in Mathematics 1875), Berlin: Springer-Verlag, 2006. |

[7] | A. Law and D. Kelton, Simulation Modeling and Analysis, 3rd ed., Boston: McGraw-Hill, 2000. |

[8] | T. Griffiths and Z. Ghahramani, “Infinite Latent Feature Models and the Indian Buffet Process,” in Proceedings of the Eighteenth Annual Conference on Neural Information Processing Systems (NIPS 18), Whistler, Canada, 2004. books.nips.cc/papers/files/nips18/NIPS2005_0130.pdf. |

[9] | R. Thibaux and M. I. Jordan, “Hierarchical Beta Processes and the Indian Buffet Process,” in Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics (AISTATS 2007), San Juan, Puerto Rico, 2007. jmlr.org/proceedings/papers/v2/thibaux07a/thibaux07a.pdf. |

[10] | N. D. Goodman. “The Principles and Practice of Probabilistic Programming,” in Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, (POPL 2013) Rome, Italy, 2013 pp. 399-402. doi:10.1145/2429069.2429117. |

[11] | N. D. Goodman, V. K. Mansinghka, D. Roy, K. Bonawitz, and J. B. Tenenbaum, “Church: A Language for Generative Models,” in Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence (UAI2008), Helsinki, Finland, 2008. www.auai.org/uai2008/UAI_camera_ready/goodman.pdf. |

[12] | I. Murray. Markov Chain Monte Carlo [video]. (Aug 8, 2013) videolectures.net/mlss09uk_murray_mcmc. |

[13] | D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge, UK: Cambridge University Press, 2003. www.inference.phy.cam.ac.uk/itila/book.html. |

[14] | C. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed, New York: Springer, 2004. |

[15] | C. Kemp, J. B. Tenenbaum, T. L. Griffiths, T. Yamada, and N. Ueda, “Learning Systems of Concepts with an Infinite Relational Model,” in Proceedings of the 21st National Conference on Artificial Intelligence (AAAI-06), Boston, MA, 2006.www.aaai.org/Papers/AAAI/2006/AAAI06-061.pdf. |

[16] | N. D. Goodman, J. B. Tenenbaum, T. J. O’Donnell, and the Church Working Group. “Probabilistic Models of Cognition.” (Aug 8, 2013) projects.csail.mit.edu/church/wiki/Probabilistic_Models_of _Cognition. |

J. Cassel, “Probabilistic Programming with Stochastic Memoization,” The Mathematica Journal, 2014. dx.doi.org/doi:10.3888/tmj.16-1. |

John Cassel works with Wolfram|Alpha, where his primary focus is knowledge representation problems. He maintains interests in real-time discovery, planning, and knowledge-representation problems in risk governance and engineering design. Cassel holds a Master of Design in Strategic Foresight and Innovation from OCADU, where he developed a novel research methodology for the risk governance of emerging technologies.

**John Cassel**

Wolfram|Alpha LLC

100 Trade Center Drive

Champaign, IL 61820-7237

*jcassel@wolfram.com*

We study well-known chocolate games and new ones that we have invented. Most of the games presented here have simple formulas for the set of losing states, but many do not, and it is interesting that those have very beautiful graphs for the set of losing states.

The setup for Nim is a set of heaps that contain objects. For example, three heaps might contain three, four, and five objects. Two players alternate in taking any positive number of objects away from a single heap. The player who takes the last object loses.

Chocolate games generalize Nim to two dimensions. In this section we study well-known chocolate games and variations created by the authors. The chocolate games are simple to play, but they are interesting mathematically.

**Definition 1**

In a rectangular piece of chocolate made of squares, the brown parts are sweet and a single blue square is very bitter. Two players play alternately. To move, a player breaks the chocolate in a straight line along one of the grooves and eats the rectangular piece broken off. The player who takes the bitter square loses. The same game can be played with a rectangular box made of cubes.

There are other types of chocolate games; one of the most well-known is Chomp [1]. Again the chocolate is a rectangle made of squares, with the top-left square being very bitter. To move, a player removes a square and all the squares below it and to its right. Many people have studied this game and many interesting theories about it have been developed.

Each chocolate game treated in this article satisfies an inequality, so they are very different from Chomp, and for certain types of inequalities the mathematical structure of the winning strategy is very simple.

Play the chocolate game in Figure 1 with “nim-sum” unchecked. Choose the values of , , and then click “new game.” Each time you click a line (not a square), you break the chocolate into two pieces along that line and eat the piece without the bitter square [2]. The game coordinates , , are the number of squares to the left, above, and to the right of the blue square, respectively.

In chocolate games there are two kinds of states.

**Definition 2**

A winning state is a state from which we can force a win as long as we play correctly at every stage. A losing state is a state from which we lose no matter how well we play, if our opponent plays correctly.

The mathematical structure of chocolate game 1 is well-known, and there is a formula to calculate losing states.

**Proposition 1**

The state is a losing state of chocolate game 1 if and only if , where is the bitwise exclusive-or operation (the same as `BitXor` in *Mathematica*).

For a proof, see [2].

Once we know the formula for losing states, the strategy to win is clear. Suppose that you have a winning state. Then you can choose a move that gives your opponent a losing state. Any move by your opponent is a winning state for you, and you can always move to a losing state again. Finally, you reach and win the game.

You can use this strategy in chocolate game 1 by selecting “nim-sum.” For each state you see , and you know if the state is a losing state or a winning state.

In the box chocolate game, the bitter (blue) cube is at the bottom of the 3D chocolate; to see it, drag to rotate.

Chocolate game 2 is a little more complicated than chocolate game 1, but the mathematical structure is almost the same.

In chocolate game 2, there are five controls. (You cannot cut the chocolate by clicking a face.)

**Proposition 2**

For a proof, see [2].

Here are some new chocolate games.

In chocolate game 3, you can cut the chocolate in three ways: to the left or right of the blue piece, or above it. The coordinates , , and are the maximum number of times you can cut the chocolate in those directions.

The geometry is such that clicking lines with low values of also deletes lines, so that the coordinates always satisfy ; that is, .

**Proposition 3**

In fact this proposition is valid for chocolate games with the inequality for even . Although Proposition 3 is very similar to Proposition 1, their proofs are very different. For proofs, see [3] and [4].

Chocolate game 4 is the same as chocolate game 3, but you can vary . In this game ; that is, for any state.

The authors are the first researchers who studied chocolate games whose coordinates satisfy inequalities.

For certain types of inequalities, the mathematical structure of the set of losing states is very similar to that of the well-known games of Nim and rectangular chocolate, but for most inequalities the mathematical structure of the set of losing states is very different.

The authors could not find any simple formula for the losing states of triangular chocolate games whose coordinates satisfy the inequality . They present a conjecture for the formulas of losing states in Section 3 of [5]. It seems that chocolate games whose coordinates satisfy for odd do not have simple formulas for the set of losing states, because the set of losing states has very complicated graphs. In Figure 9 you can see graphs made by the set of losing states; for even the graph is a Sierpinski gasket and for odd the graph is complicated. As for the graphs made by losing states, see Figure 6.

**Conjecture 1**

For a triangular chocolate game satisfying , is a losing state if and only if when (mod 4), and is a losing state if and only if when (mod 4).

This conjecture is Proposition 4 when and is proved for in [6].

For other types of chocolate games that the authors studied, see [7], [6], [8], and [9].

**Proposition 4**

For the triangular chocolate game that satisfies the inequality , is a losing state if and only if .

For a proof, see [10].

**Conjecture 2**

For the chocolate game that satisfies the inequalities and , the state is a losing state of this chocolate game if and only if .

The authors have proved but not published this proposition.

Here the authors present a *Mathematica* program to calculate losing states of the chocolate game whose coordinates satisfy the inequality . First define the function for each state of the chocolate and the function , where is a set of positive integers.

**Definition 3**

**Definition 4**

The function is the smallest non-negative integer that is not an element of the set (the “minimum excluded ordinal”).

*Example 1*

and .

This defines the Grundy number recursively.

**Definition 5**

**Proposition 5**

This is a well-known fact in the field of combinatorial games.

This game has two coordinates with the inequality ; and are the height and width of the chocolate when we ignore the blue part.

See Figure 8; here is the upper bound of the number of the rows and columns you can remove. Below the game is the table of Grundy numbers .

**Conjecture 3**

- , for any natural number .
- If , then .

By Proposition 5, is a losing state if and only if the Grundy number .

Here is a *Mathematica* program to calculate losing states using Grundy numbers. Here . This *Mathematica* program comes from [11]. The definition for `mex` is as in the code for Figure 8. The definitions for `move3` and `Gr3` are modifications of `move` and `Gr` from that code, so we change their names slightly.

These are the candidate states for a particular case.

This calculates the losing states using Grundy numbers.

Indeed these are losing states.

There are no other losing states.

Here the authors define the Grundy-like numbers , which are efficient to calculate.

**Definition 6**

- If , then let . If , then let . If , then let .
- Suppose that and .
- Suppose that and .
- Suppose that and .

**Proposition 6**

For a proof of this proposition see Theorem 4.2 of [4].

This defines the corresponding *Mathematica* function `G2`.

This calculates the losing states using Grundy-like numbers.

They are exactly all the losing states.

Here are some graphs of the set of losing states of chocolate games. (For more, see [10].)

By Proposition 2 there is a simple formula for the set of losing states of the chocolate games satisfying , where is even. In Figure 9 the graph is a Sierpinski gasket for even .

When is odd, the graph is very different from a Sierpinski gasket. Since the graph is quite complicated, it seems likely that there is no simple formula for the set of losing states. As for the relation between the graphs of losing states and the Sierpinski gasket, see [7] (in Japanese).

Figure 10 shows that the graph is the Sierpinski gasket for odd , so it seems that there may be a simple formula for the set of losing states in that case.

When , by Proposition 3 there is a simple formula for the set of losing states. When , the state is a losing state of this chocolate game if and only if [6]. The authors are now trying to find a simple formula for the set of losing states for an arbitrary odd number .

When is even, the graph is very different from the Sierpinski gasket. Since the graph is quite complicated, it seems that there is no simple formula for the set of losing states.

We can study many kinds of inequalities. For example, the chocolate games that satisfy the inequalities or are interesting, when is even and is a non-negative integer. The authors are studying these games.

Dr. Miyadera taught the rectangular chocolate games presented in [2] to his students and encouraged them to make new chocolate games. Students were very good at proposing new types of chocolate games, including the ones in Figures 3-8. These students were the first people who studied chocolate games that satisfy inequalities.

As for the triangular chocolate game applications on smart phones, see [12], [13], and [14].

The fact that high school students could discover new theorems using computer algebra systems is very important for mathematics education. There are many teachers around the world who make students rediscover formulas in the classroom. In so-called “learning by doing,” students are supposed to rediscover formulas and theorems, while teachers know these formulas and theorems. In the authors’ high school research projects, students discover new things! The difference between discovery and rediscovery is very big, and the method the authors introduced can change mathematics education greatly.

The method used in the group is as follows.

First Dr. Ryohei Miyadera introduces some problems to the high school students, and after that students are required to create new problems by changing the conditions of the original problems. Sometimes they make a problem that is a little bit different from the original problem. At other times they propose a completely new problem. When the proposed problem seems to be very promising to Dr. Miyadera, he and his students begin to research it. They make computer programs using *Mathematica *and try to find interesting facts by calculation. Once they discover a new fact, they begin to prove it. Even if the proposed problem does not seem to be interesting enough, Dr. Miyadera encourages students to study it. It is often the case that a problem that seems to be trivial to him turns out to be a good problem. Some students have a kind of intuition that is beyond the imagination of Dr. Miyadera, who is an active mathematician himself. This method of research looks simple, but it is very effective.

In the history of this research group, more than 60 students have participated. Dr. Miyadera discovered that some students are very good at proposing new ideas while others are good at proving formulas and theorems. According to some researchers of mathematics education, there has never been a group of high school students that have been constantly discovering new formulas and theorems.

This group of students won the first prize in the Canada Wide Virtual Science Fair 2007, 2008, 2009, 2010, 2011, 2012, and 2013. They won the Imai Isao Award in 2007, which is the first prize in the Japan Wide High School Research Contest supported by Kogakuin University in Japan. They won the first prize in the Japan Science & Engineering Challenge 2008, and represented Japan in the International Science and Engineering Fair. (Japan declined to send them to the fair because of the epidemic of swine influenza.) They became finalists in the Japan Science & Engineering Challenge in 2011, 2012, and 2013. They became Google Science Fair Regional Finalists in 2012 and became semifinalists in the Yau High School Mathematics Awards in 2012 and 2013.

For a detailed explanation about the authors’ high school mathematics research project, see [15], [16], [17], [18], [19], [20], and [21]. The authors have presented their research at the International *Mathematica* Symposium in [22], [23], and [24].

They have also presented their research in the *Mathematica* Demonstrations Project, and Ryohei Miyadera was selected as a featured contributor. See [25], [26], and [27].

Dr. Miyadera received the Excellent Teacher Award from the Ministry of Education of Japan in 2012, and he received the Wolfram Innovator Award in 2012 for his research with high school students.

If you are interested in doing a high school or undergraduate mathematics research project, one of the best books to use is by Cowen and Kennedy [11].

If students use *Mathematica* freely with an experienced mathematician, they can discover a lot of new and interesting facts of mathematics. The difference between a true discovery and rediscovery is great. We think that the best way to learn how to be creative is to create new and interesting things.

We would like to thank Professor Robert Cowen of the City University of New York for his encouragement when we met [22]; he gave his book [11] to us. We learned how to use *Mathematica* in our research by using his book. We also would like to thank Ed Pegg Jr of Wolfram Research and Professor Tadashi Takahashi for giving us valuable advice.

[1] | Wikipedia. “Chomp.” (Dec 4, 2013) en.wikipedia.org/wiki/Chomp. |

[2] | A. C. Robin, “A Poisoned Chocolate Problem,” Problem Corner, The Mathematical Gazette, 73(466), 1989 pp. 341-343. (An answer for the above problem is in 74(468), 1990pp. 171-173.) |

[3] | R. Miyadera, S. Nakamura, and R. Hanafusa, “New Chocolate Games—Variants of the Game of Nim,” Proceedings of the Annual International Conference on Computational Mathematics, Computational Geometry & Statistics, Singapore, 2012 pp. 122-128.dl4.globalstf.org/?wpsc-product=new-chocolate-game-variants-of-the-game-of-nim. |

[4] | R. Miyadera, S. Nakamura, and R. Hanafusa, “New Chocolate Games,” GSTF Journal of Mathematics, Statistics and Operations Research, 1(1), 2012 pp. 111-116.dl4.globalstf.org/?wpsc-product=new-chocolate-games. |

[5] | S. Nakamura, Y. Naito, R. Miyadera et al., “Chocolate Games That Are Variants of Nim and Interesting Graphs Made by These Games,” Visual Mathematics, 14(2), 2012. www.mi.sanu.ac.rs/vismath/miyaderasept2012/index.html. |

[6] | T. Yamauchi, T. Inoue, and Y. Tomari, “Variants of the Game of Nim That Have Inequalities as Conditions,” Rose-Hulman Institute of Technology, Undergraduate Math Journal, 10(2), 2009. www.rose-hulman.edu/mathjournal/archives/2009/vol10-n2/paper12/v10n2-12pd.pdf. |

[7] | R. Miyadera and S. Nakamura, “A Chocolate Game,” Mathematics of Puzzles, Journal of Information Processing (special issue), 53(6), 2012 pp. 1582-1591 (in Japanese). |

[8] | M. Naito, T. Yamauchi, T. Inoue et al., “Discrete Mathematics and Computer Algebra System,” Proceedings of the 9th Asian Symposium on Computer Mathematics-Mathematical Aspects of Computer and Information Sciences Joint Conference (ASCM-MACIS), Fukuoka, Japan: Kyushu University, 2009 pp. 127-136.gcoe-mi.jp/temp/publish/a2019bae287f4628e4bfc9d273150000.pdf. |

[9] | MathPuzzle.com. “The Bitter Chocolate Problem.” Material added 8 Jan 06 (Happy New Year). www.mathpuzzle.com/26Feb2006.html. |

[10] | T. Nakaoka, R. Miyadera et al., “Combinatorial Games and Beautiful Graphs Produced by Them,” Visual Mathematics, 11(3), 2009. www.mi.sanu.ac.rs/vismath/miyaderasept2009/index.html. |

[11] | R. Cowen and J. Kennedy, Discovering Mathematics with Mathematica, Erudition Books, 2001. |

[12] | M. Fukui, “Bitter Chocolate Games 1.” itunes.apple.com/jp/app/bitter-chocolate-games-1/id495880660?mt=8. |

[13] | M. Fukui, “Bitter Chocolate Games 2.” itunes.apple.com/jp/app/bitter-chocolate-games-2/id502476677?mt=8. |

[14] | S. Nakamura, “Poison Chocolate Game.” market.android.com/details?id=com.kgshmathclub.chocoapp. |

[15] | R. Miyadera, T. Inoue, and S. Nakamura, “High School Mathematics Research Project Using Computer Algebra System,” Proceedings of the 16th International Seminar on Education of Gifted Students in Mathematics, The Korean Society of Mathematical Education, Chungnam National University, Daejeon, Korea, 2011 pp. 303-316. |

[16] | S. Hashiba, D. Minematsu, H. Shimoda, and R. Miyadera, “How High School Students Can Discover Original Ideas of Mathematics Using Mathematica,” Mathematica in Education and Research 11(3), 2006. |

[17] | H. Matsui, D. Minematsu, T. Yamauchi, and R. Miyadera, “Pascal-Like Triangles and Fibonacci-Like Sequences,” Mathematical Gazette, 94(529), 2010 pp. 27-41. |

[18] | R. Miyadera, T. Hashiba, Y. Nakagawa, T. Yamauchi, H. Matsui, S. Hashiba, D. Minematsu, and M. Sakaguchi, “Pascal-Like Triangles and Sierpinski-Like Gaskets,” Visual Mathematics, 9(1), 2007. www.mi.sanu.ac.rs/vismath/miyad/pascalliketriangle.html. |

[19] | H. Matsui and T. Yamauchi, “Formulas for Fibonacci-Like Sequences Produced by Pascal-Like Triangles,” Rose-Hulman Institute of Technology Undergraduate Math Journal, 9(2), 2008. www.rose-hulman.edu/mathjournal/archives/2008/vol9-n2/paper11/v9n2-11p.pdf. |

[20] | H. Matsui, N. Saita, K. Kawata, Y. Sakurama, and R. Miyadera, “Elementary Problems, B-1019,” Fibonacci Quarterly, 44(3), 2006. www.fq.math.ca/Problems/August2006elementary.pdf. |

[21] | R. Miyadera, “General Theory of Russian Roulette.” library.wolfram.com/infocenter/MathSource/5710. |

[22] | R. Miyadera, K. Fujii et al., “How High School Students Could Present Original Math Research Using Mathematica,” The International Mathematica Symposium, 2003. library.wolfram.com/infocenter/Conferences/4983. |

[23] | R. Miyadera and D. Minematsu, “Curious Properties of an Iterative Process: How Could High School Students Present Original Mathematics Research Using Mathematica?,” The International Mathematica Symposium, 2004. library.wolfram.com/infocenter/Conferences/6055. |

[24] | R. Miyadera, “Using Mathematica in Doing the Research of Mathematics with High School Students,” Proceedings of the Fourth International Mathematica Symposium (IMS2001), Tokyo, 2001. |

[25] | H. Matsui, T. Yamauchi, D. Minematsu, and R. Miyadera. “Pascal-Like Triangles Made from a Game” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/PascalLikeTrianglesMadeFromAGame. |

[26] | H. Matsui, T. Yamauchi, D. Minematsu, and R. Miyadera. “Pascal-Like Triangles Mod ” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/PascalLikeTrianglesModK. |

[27] | T. Inoue, T. Nakaoka, and R. Miyadera. “A Bitter Chocolate Problem That Satisfies an Inequality” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ABitterChocolateProblemThatSatisfiesAnInequality. |

R. Miyadera, S. Nakamura, Y. Okada, T. Ishikawa, and R. Hanafusa, “Chocolate Games,” The Mathematica Journal, 2013. dx.doi.org/doi:10.3888/tmj.15-12. |

Ryohei Miyadera received a Ph.D. (mathematics) at Osaka City University and received a second Ph.D. (mathematics education) at Kobe University. He has two fields of research: probability theory of functions with values in an abstract space and applications of *Mathematica* to discrete mathematics. He and his high school students have been doing research in discrete mathematics for more than 15 years. They have talked at IMS 2001 [24], IMS 2003 [22], and IMS 2004 [23]. They have also published more than 22 refereed articles. Ryohei Miyadera received the Excellent Teacher Award from the Ministry of Education of Japan in 2012, and he received the Wolfram Innovator Award in 2012. His hobby is long-distance running, and he is one of the best runners in the over-55-years-old group of his prefecture.

Shunsuke Nakamura has graduated from Kwansei Gakuin High School and is preparing for the entrance examination to university. He did mathematical research with Dr. Miyadera when he was a high school student at Kwansei Gakuin. He has published papers in [3], [4], [5], and [7]. He is a member of the Information Processing Society of Japan. Nakamura’s hobbies are playing the piano and playing video games.

Yu Okada is a high school student. He is a very serious player of Massively Multiplayer Online RPGs.

Tomoki Ishikawa is a university student at Kwansei Gakuin. He did mathematical research with Dr. Miyadera when he was a high school student at Kwansei Gakuin. He has published papers on origami mathematics. He is a photographer who received an excellent photography award in the Japan Wide Cultural Festival.

Ryo Hanafusa is a university student at Kwansei Gakuin. He did mathematical research with Dr. Miyadera when he was a high school student at Kwansei Gakuin. He has published papers in [3] and [4]. He is a photographer who received an excellent photography award in the Japan Wide Cultural Festival.

**Ryohei Miyadera**

*Mathematics Department
Kwansei Gakuin High School*

**Shunsuke Nakamura**

*Kwansei Gakuin High School*

*nakashun1994@gmail.com*

**Yu Okada**

*Kwansei Gakuin High School*

*alm1_a1m_aim@ezweb.ne.jp*

**Tomoki Ishikawa**

*Kwansei Gakuin High School*

* tom94826@gmail.com*

**Ryo Hanafusa**

*Kwansei Gakuin High School*

*ryo3waygate@yahoo.co.jp*

Affymetrix gene expression microarrays (“chips”) are a commercial implementation of a powerful concept originally introduced to the world by Shena and colleagues in 1995 [1]. When successfully implemented, gene expression microarrays let a biologist measure the expression of thousands of genes simultaneously in a biological sample, such as heart tissue, and further, to compare that measure of expression between two biological states, such as diseased heart tissue and healthy heart tissue. In many ways, the introduction of microarray technology has created a revolution in biology, transforming the field into a “big data” science like its sister disciplines of physics and chemistry.

Gene expression microarrays (Figure 1A) are manufactured by attaching strands of deoxyribonucleic acid (DNA), corresponding to different genes of an organism, across the surface of a glass slide. The basic process of identifying genes that are expressed begins with the extraction of messenger RNA (mRNA) from a source, for example, healthy heart cells (Figure 1B). The molecule mRNA is made by cells when a gene is expressed, meaning its physical presence—assuming it can be reliably detected—is an indicator of gene expression. By fluorescently labeling the mRNA and hybridizing it to the surface of the chip, it is possible to quantify the intensity of multiple genes’ expression from that biological source by using a scanner able to measure a fluorescent signal. When this process is repeated on a different biological source, such as diseased heart tissue, a separate gene expression profile is created for the diseased tissue, which can then be computationally compared to the expression profile from the healthy tissue. In this manner, genes that are more highly expressed or more highly repressed in diseased tissue can be identified by comparing their expression profile to the expression profile of the same genes in the healthy tissue. This has obvious implications for determining which genes may be playing a role in disease development or any other biological process of interest, from cancer metastasis and drug resistance in medicine to fruit ripening and drought resistance in agriculture.

**Figure 1.** (A) Basic microarray design and layout. (B) Extraction of mRNA—expressed from the nucleus of a healthy heart cell—and its subsequent fluorescent labeling. (C) Hybridization of the fluorescently labeled mRNA (“target”) to the surface of an Affymetrix microarray chip, and its subsequent scanning to quantify the fluorescent signal, assumed to be proportional to gene expression.

The Affymetrix implementation of gene expression microarrays utilizes probesets, synthesized in place, on the surface of the microarray chip (Figure 1C). Probesets are groups of small DNA fragments that are complementary to different regions of the same mRNA molecule made whenever a gene is expressed. By combining the fluorescent signal of the probeset group, a single measure of gene expression is arrived at computationally, which is the primary focus of the algorithm presented here. Probesets are composed of groups of perfect match (“PM”) and mismatch (“MM”) probes. In the context of microarray analysis, the term “probe” refers to the strands of DNA physically tethered to the microarray chip and the term “target” refers to the fluorescently labeled sample of mRNA obtained from a biological source, which will be hybridized to the probes to measure gene expression. Perfect match probes are single-strand sequences of DNA, usually 25 nucleotides in length, which have perfect complementarity to the mRNA sequence to which they are designed to hybridize. Mismatch probes are identical to PM probes, with the exception of one nucleotide in the center of the molecule (typically at position 13) that is not a proper match to the mRNA with which it is designed to hybridize. The purpose of MM probes is to measure background fluorescent signal off the surface of the chip, which is one source of technical noise.

The analysis of microarray data involves numerous steps, some of which are not universally performed, but whose combination is collectively referred to as an “analysis pathway.” The steps in an analysis pathway typically involve:

- background correction: performed to remove fluorescent signal not due to biology
- probe normalization: used to place the individual datasets of an experiment on the same scale, so the datasets can be compared accurately
- perfect match (PM) probe correction: used to correct biases in the PM probe signal, often due to differences in DNA sequence between the probes
- summarization: performed to obtain a single measure of gene expression from the multiple measurements obtained by each probeset; this process often attempts to correct “probe” and “chip” technical noise
- probeset normalization: sometimes performed to make the probesets between datasets more directly comparable
- differentially expressed genes (DEG) test: uses a statistical test to identify “true” differentially expressed genes

Despite all of its positive aspects, microarray technology requires considerable knowledge to use effectively, as the raw signal that is generated from the technology is almost always noisy. The scientific literature is rife with algorithms designed to remove various sources of known error, and it is immediately clear that there is no “perfect” algorithm that is universally useful in all experimental situations. Even so, a seminal article by Zhu and colleagues [2] evaluated different combinations of commonly used algorithms—representing over 40,000 different analysis pathways—using a precisely controlled “spike-in” dataset, which allowed the researchers to identify the most important steps common to a good analysis pathway.

The algorithm presented here represents a merging of several of the “best step” analysis pathway practices identified in [2]. Specifically, the algorithm presented here uses the following analysis pathway:

- background correction: none
- probe normalization: performed using quantile normalization [3]
- perfect match (PM) probe correction: none
- summarization: performed using median polish [4]
- probeset normalization: none
- differentially expressed genes (DEG) test: while probability data is provided to aid interpretation, the identification of differentially expressed genes is not performed with statistical methods, but instead relies on graphical interpretation of the processed data

The algorithm presented here does not perform background correction, perfect match probe correction, or probeset normalization because the evidence presented in [2] suggests that these steps are at best unnecessary and sometimes even detrimental. Readers interested in a deeper discussion of microarray technology and data analysis are referred to the excellent reviews in [5, 6].

The AffyDGED algorithm is template-driven, meaning that the algorithm expects several pieces of user-defined information to be provided in a notebook cell used as a template for entering the information.

The example above contains several variables that must be completed by the user to let AffyDGED do its job properly.

The variables requiring user input are:

- cellocation: This variable holds the directory location for finding the CEL data of the microarray hybridizations. CEL files contain the raw fluorescent data from a microarray experiment using Affymetrix technology.
`affycdflocation`: This variable holds the directory location for finding the Affymetrix CDF library file. CDF stands for “chip description file” and refers to an Affymetrix file that describes, among other things, the location of the probes on the specific type of Affymetrix chip being used.`savelocationroot`: This variable holds the location where the user would like the final results of the analysis to be saved.`qnormversion`: This variable lets the user select between two options. The option “all” can be entered to perform quantile normalization using all the chips involved in an experiment at once, or alternatively, the option “condition” can be entered, which directs the algorithm to perform quantile normalization by condition, that is, perform quantile normalization twice, once using the data in the experimental condition (such as diseased heart tissue) and then a second time using the control condition data (such as healthy heart tissue). When in doubt, it is recommended that “all” be used.`studyname`: This variable lets the user name the experiment/study being processed by the AffyDGED algorithm. The output of AffyDGED is saved using this name to the location provided in “savelocationroot” above.`experimentchips`: This variable contains a list of the experimental condition datasets being studied.`controlchips`: This variable contains a list of the control condition datasets being studied.

To illustrate the features of the AffyDGED algorithm, we use data from a modestly sized microarray experiment involving the detection of differentially expressed genes between the saliva of pancreatic patients and healthy individuals [7]. All microarray data used in this study and presented here is publicly available at NCBI’s Gene Expression Omnibus portal (www.ncbi.nlm.nih.gov/geo), using the access number GSE14245.

The first tasks completed by AffyDGED include the loading of raw data, the determination of the physical dimensions of the chip data, and the conversion of Affymetrix probe position coordinates to equivalent *Mathematica* indices.

The `chipdimensions` and `affyindextoMMAindices` modules are designed to establish the number of rows and columns of probe data on the microarray chips, as well as to convert the probe position coordinates as assigned by Affymetrix to equivalent *Mathematica* indices.

For example, the chips used here (Human Genome U133 Plus 2.0) happen to be square, with 1,164 rows of information and 1,164 columns of information.

Affymetrix uses a single-number index (present on the Affymetrix CDF file) created from coordinates of the probes present on their microarray chips. The single-number index is derived from a formula used by Affymetrix that assumes an index of refers to the uppermost-leftmost position of the chip. To successfully load the probeset data into usable groups in *Mathematica*, it is necessary to shift the coordinates used by Affymetrix into indexing used by *Mathematica*.

Here is an example of the data contained within the Affymetrix CDF file.

The AffyDGED algorithm parses out the single-number indices for the perfect match probes of each probeset and converts that positional information into usable indices for *Mathematica*. The mismatch probes are purposely ignored in the AffyDGED algorithm because they often produce signals higher than the perfect match probes, which is an indication that the mismatch probes are not performing as originally intended by Affymetrix engineers.

There are 11 individual numbers, referring to the positions (in Affymetrix coordinates) of 11 perfect match probes of a single probeset used to measure the expression of a specific gene.

Here is the same positional information, now expressed in *Mathematica*’s indexing system.

There are now 11 groups of *Mathematica* indices referring to the same data and accessible by conventional *Mathematica* indexing.

We can now take a look at the raw data from a single chip used in the pancreatic cancer study that has been loaded. Notice the extreme range of values typically seen in microarray experiments.

For this reason, we look at a histogram of the data using a logarithmic scale (Figure 2).

**Figure 2.** A histogram of the raw fluorescence intensity data (log scale) contained in the microarray chip GSM356796, used to measure the expression of genes in the saliva of a pancreatic cancer patient.

The majority of the data has an approximately Gaussian appearance (on a log scale), while still harboring extreme values on the rightward tail (look carefully along the axis). This shape is very characteristic of microarray data.

The next steps in processing include transforming the raw data to a scale, determining the probeset size specific for the chip in use, and performing quantile normalization.

The convention of transforming raw microarray data by is almost universally used in the microarray community, because it performs two useful functions. First, it makes the distribution of raw data more Gaussian (although certainly not perfectly so), and it aids interpretation of gene expression ratios for the end user, because it is easier to appreciate that a ratio of and on a log scale indicates the same degree of “up” and “down” regulation for a gene, as opposed to and on an absolute scale.

In the example shown here, this is the probeset size (the number of probes making up each probeset).

Quantile normalization is performed to place each dataset of the experiment on a common scale so the datasets can be appropriately compared to each other.

Using `BoxWhiskerChart`, this shows the difference between the pre-quantile normalized data and the post-quantile normalized data (Figure 3).

**Figure 3.** A box-and-whisker comparison of all 24 microarray chips used to compare gene expression between the saliva of pancreatic cancer and healthy patients, before and after quantile normalization.

Following quantile normalization, the probesets are summarized (i.e., a single measure of gene expression is generated) by first performing median polish and then taking the mean of the polished values for all probes within a probeset. After the probesets are summarized, the differential expression of each gene is obtained by subtracting the expression of a gene in the control condition from the expression of the gene in the experimental condition. For example, if the expression of a gene in the saliva of pancreatic cancer patients (the “experimental” condition) is 2.1 and the expression of the same gene in the saliva of healthy patients (the “control” condition) is 1.4, then the differential expression of the gene is .

Here are the distributions of gene expression in the saliva of pancreatic cancer and healthy patients, as well as the distribution of differentially expressed genes between the two states (Figure 4).

**Figure 4.** Histograms of the summarized (post median polish) gene expression values contained in the saliva of pancreatic cancer and healthy patients, as well as the difference in gene expression between those two biological groups.

From these results, simulations are performed to calculate probability -values, which answer the question, If we were to repeat this experiment many times, under the same experimental conditions as this study, how often would we find results as (or more) extreme than we have observed for each gene in the current study? The simulations performed to calculate the -values use the corrected fluorescent signal values contained within the experimental and control datasets and thus make the assumptions that the corrections are valid and that the data is now a good representative of the biological “truth” being studied. If these assumptions are correct, the probability values reported help the end user to gauge how rare a gene’s measure of differential expression is, but not—as is traditionally used in the microarray community—to make a decision about statistical or biological significance.

Following this, the algorithm organizes the output into a more human-readable table.

Here is a random sample of our current results.

The output columns are as follows:

Column 1: the signal-corrected fluorescence intensity of gene expression in the experimental condition

Column 2: the signal-corrected fluorescence intensity of gene expression in the control condition

Column 3: the measure of differential expression, obtained by subtracting column 2 from column 1

Column 4: the -value obtained through simulation

Column 5: the probeset name as assigned by Affymetrix

Column 6: descriptive information for the probeset including the genbank accession number, gene name, and gene product information

Due to the length of descriptive data in column 6, the output here has been purposefully rearranged to print column 6 underneath each data point’s first five column entries.

The final steps in processing the data include establishing the upper and lower thresholds for determining when a gene is considered up-regulated (turned “on”) in the experimental condition versus the control condition, and when a gene is considered down-regulated (turned “off”). Further, the algorithm saves the final results and provides a summary report of the completed analysis.

AffyDGED saves all the data and a list of differentially expressed genes to the location defined by the user in the template above. Further, both lists of data are saved in two formats, one convenient for users to continue to explore the data in *Mathematica* and the other conveniently readable in Microsoft Excel.

Affymetrix gene expression array technology utilizes a suite of library files containing important annotation information about the layout and content of its microarray products. AffyDGED can analyze any Affymetrix gene expression experiment as long as the .cdf (chip description file) and .gin (gene information) library files associated with the specific type of microarray chip being used are provided. These files are freely available to the public at www.affymetrix.com/support/technical/libraryfilesmain.affx. The variable “affycdflocation”, defined in the user template, holds the directory location of where the user has stored the .cdf file, and AffyDGED expects the .gin file to also be stored in the same location. Having the .cdf and .gin files together happens naturally, as they are packaged together by Affymetrix and unzip to the same location upon download.

As mentioned previously, there is no universally correct algorithm for processing microarray data in all experimental circumstances. AffyDGED was developed under the guidance of [2] because the exact expression of genes in this study was precisely controlled and therefore is a useful gauge of how effectively a microarray analysis algorithm is performing.

Supplemental file 5 in [2] describes the 1,944 differentially expressed genes and 3,426 non-differentially expressed genes that were purposefully “spiked-in” to the microarray experiment. When this dataset is analyzed by the AffyDGED algorithm described here, the thresholds for determining “up” and “down” gene expression are calculated to be 0.225 and , respectively (the red lines in Figure 5). Establishing the thresholds for determining differential expression relies on the observation illustrated in Figure 5, that a plot of the processed data always reveals a tight clustering of data about the line . As the reader scans above and below that axis, note how the density of data noticeably separates from the cluster along that line. This observation was used to develop code that scans vertically up and down in small increments and establishes a breakpoint in each direction any time the density of data at a vertical position is 50% less than it was at the previous increment. These breakpoints become the thresholds for determining differentially expressed up and down genes.

**Figure 5.** The resulting differential expression threshold determination plot by AffyDGED when processing the data in [2] (accession number GSE21344).

The AffyDGED algorithm identifies 1,832 genes as differentially expressed in [2]. Of these, 1,591 genes overlap with the true 1,944 differentially expressed genes for a true positive rate of . This means AffyDGED was unable to correctly identify 18% of the true list of differentially expressed genes. While perhaps surprising to readers unfamiliar with microarray analysis, this places AffyDGED’s performance among the top performers of leading algorithms identified by [2]. State of the art at this time in microarray analysis means accepting a 1 out of 5 “miscall” rate in differential expression detection. Because of the complexity of microarray technology and of all the steps that occur prior to the actual algorithmic analysis of the data, it is important to realize that at least some of the miscalls by any microarray algorithm are really due to factors that are poorly controlled for by the underlying technology, and do not indicate a weakness in the algorithm itself [8].

To gauge the performance of AffyDGED, several publicly available datasets of different sizes and complexity were profiled. The first row of Table 1 shows the series accession number for each dataset available at NCBI’s Gene Expression Omnibus. All timings were obtained using quantile normalization with all chips (i.e. qnormversion set to “all”). Timings were acquired running *Mathematica* 9.0 under Windows 7 (64 bit) using an Intel Core i5-2500K processor overclocked to 4.48 Ghz.

AffyDGED performs very well, in all cases completing its analysis in less than seven and a half minutes. The largest chip in this comparison is the Human chip, where each of the processed files contains 13.2 Mb of data. AffyDGED is able to process the combined worth of data in a practically usable time frame.

Microarray technology continues to be heavily used by the biomedical and basic science research communities throughout the world. AffyDGED brings a contemporary algorithm useful in the real world to the *Mathematica* user community interested in exploring fundamental biology questions with their favorite computational tool chest.

[1] | M. Schena, D. Shalon, R. W. Davis, and P. O. Brown, “Quantitative Monitoring of Gene Expression Patterns with a Complementary DNA Microarray,” Science, 270(5235), 1995 pp. 467-470. www.jstor.org/stable/2889064. |

[2] | Q. Zhu, J. C. Miecznikowsk, and M. S. Halfon, “Preferred Analysis Methods for Affymetrix GeneChips. II. An Expanded, Balanced, Wholly-Defined Spike-in Dataset,” BMC Bioinformatics, 11(285), 2010. doi:10.1186/1471-2105-11-285. |

[3] | B. M. Bolstad, R. A. Irizarry, M. Ästrand, and T. P. Speed, “A Comparison of Normalization Methods for High Density Oligonucleotide Array Data Based on Variance and Bias,” Bioinformatics, 19(2), 2003 pp. 185-193. doi:10.1093/bioinformatics/19.2.185. |

[4] | J. W. Tukey, Exploratory Data Analysis, Reading, MA: Addison-Wesley, 1977. |

[5] | C. A. Harrington, C. Rosenow, and J. Retief, “Monitoring Gene Expression Using DNA Microarrays,” Current Opinion in Microbiology, 3(3), 2000 pp. 285-291.doi:10.1016/S1369-5274(00)00091-6. |

[6] | J. Lovén, D. A. Orlando, A. A. Sigova, C. Y. Lin, P. B. Rahl, C. B. Burge, D. L. Levens, T. I. Lee, and R. A. Young, “Revisiting Global Gene Expression Analysis,” Cell, 151(3), 2012 pp. 476-482. doi:10.1016/j.cell.2012.10.012. |

[7] | L. Zhang, J. J. Farrell, H. Zhou, D. Elashoff, D. Akin, N.-H. Park, D. Chia, and D. T. Wong, “Salivary Transcriptomic Biomarkers for Detection of Resectable Pancreatic Cancer,” Gastroenterology, 138(3), 2010 pp. 949-957.e7. doi:10.1053/j.gastro.2009.11.010. |

[8] | R. A. Rubin, “A First Principles Approach to Differential Expression in Microarray Data Analysis,” BMC Bioinformatics, 10(292), 2009. doi:10.1186/1471-2105-10-292. |

T. Allen, “Detecting Differential Gene Expression Using Affymetrix Microarrays,” The Mathematica Journal, 2013. dx.doi.org/doi:10.3888/tmj.15-11. |

Todd Allen is an associate professor of biology at HACC-Lancaster. His interest in computational biology using *Mathematica* took shape during his postdoctoral research years at the University of Maryland, where he developed a custom cDNA microarray chip to study gene expression changes in the fungal pathogen *Cryphonectria parasitica*.

**Todd D. Allen, Ph.D.**

*Harrisburg Area Community College (Lancaster Campus)
East 206 R
1641 Old Philadelphia Pike
Lancaster, PA 17602*

*Mathematica*’s industrial-strength Boolean computation capability is not used as often as it should be. There probably are several reasons for this lack of use, but it is our view that a primary reason is lack of experience in expressing mathematical problems in the form required for Boolean computation. We look at a typical problem that is susceptible to Boolean analysis and show how to translate it so that it can be tested for satisfiability with *Mathematica*’s built-in function `SatisfiableQ`. The problems we investigate come from an area of mathematics called Ramsey theory. Although Ramsey theory has been studied extensively for over 80 years and still provides many challenges, we neglect the theory (for the most part) and instead concentrate on translating the problems so that they are amenable to Boolean computation and then see what can be accomplished by computation alone. Those interested in learning a little more about Ramsey theory can consult [1]; for a standard reference, see [2].

We only concern ourselves with Boolean formulas in *conjunctive normal form* (*cnf*). A cnf is a conjunction of disjunctions of statements or their negations; the disjuncts are often called *clauses*.

An example of a cnf is ; letters represent statements and the symbols , , and stand for “or,” “and,” and “not.” A propositional formula is *satisfiable *if there is an assignment of *true* and *false* to its statements that makes the formula *true* when evaluated using the usual truth table rules.

Before *Mathematica* can test whether this formula is satisfiable using `SatisfiableQ`, we must replace , , and by `||`, `&&`, and `!`. Here is the translation to a *Mathematica* expression.

It turns out that the formula is satisfiable.

A well-known problem states that at any party with at least six people, there are at least three mutual acquaintances (each knows the other two) or three mutual strangers (each does not know the other two). In the language of graph theory, if the edges of the complete graph on six vertices, are colored red or blue, there must be either a red or a blue triangle.

We translate this into a satisfiability problem in propositional logic and then use `SatisfiableQ` to prove this result. More precisely, we show that it is not possible to color the edges of either red or blue without forming either a red or blue triangle, by building a cnf whose satisfaction is equivalent to the existence of such a coloring and then showing that this cnf cannot be satisfied.

More generally, Ramsey theory considers problems of this form: given a complete graph and integers , , with , is it possible to color the graph’s edges red or blue without obtaining a red or a blue as a subgraph?

Begin by numbering the vertices of from 1 to 6 and name its edges with ordered pairs of vertex numbers, , . For each such pair, generate two propositional variables, and , which intuitively express coloring the edge red or blue.

The cnf needs two sets of Boolean clauses.

- For each edge , the clauses and express that the edge is either red or blue, but not both.
- For each triangle , , , the clauses and express that not all edges of the triangle can be the same color.

If the cnf that is the conjunction of all these clauses is satisfiable, then it is possible to color the edges of red or blue without obtaining either a red or blue triangle. Moreover, any truth value assignment satisfying this cnf would lead immediately to a coloring of the edges by coloring the edge blue exactly when is assigned to be true. Conversely, a red-blue coloring of the edges of with no monochromatic triangle leads directly to a satisfying assignment of ; simply assign to be true if and only if the edge is colored blue, and so on. We will show that the cnf is unsatisfiable.

The function `ColorEdges` generates the first set of clauses, where the *Mathematica* expressions and play the role of and . The function states that the edges of are either one or the other of the given colors. It is generalized to three colors in the last section.

Here is the first set of clauses for the party problem.

The function `NoCompleteSubgraph` can generate the second set of clauses. gives `True` if does not contain a with all edges of the given color.

The functions `ColorEdges` and `NoCompleteSubgraph` can be extended to treat Ramsey problems for complete graphs with any number of colors and complete subgraphs.

The function puts it all together to obtain our test formula.

Here is an example.

The Boolean formula tests the party problem.

Thus, it is not possible to color the edges of red or blue and avoid a monochromatic triangle.

Although `RamseyTest` is very fast, there are ways to speed it up. (This is unimportant so far but becomes crucial when considering much larger problems!)

Since vertex 1 is connected to five other vertices, at least three of the connecting edges must be the same color; for if there were at most two red and at most two blue edges there would be at most four connecting edges instead of five. Therefore assume that the first three edges are red and not blue; by symmetry, it makes no difference which color we choose. (Replacements by one-element or unit clauses often result in significantly faster running times in automated theorem provers.)

Of course, the length of is , which is .

The edges of can be colored blue or red without any monochromatic triangles. This is easily shown by hand or with the following computation.

In fact, *Mathematica* can suggest a coloring. First generate the list `v` of propositional variables in . Then use `SatisfiabilityInstances` to indicate the coloring.

This draws the suggested edge-colored graph.

It turns out there are always at least two monochromatic triangles in [3, 4]! Since there is at least one monochromatic triangle in , assume it is formed by the edges , , and and, by symmetry, assume that its edges are all red. The next command modifies accordingly. The first replacement drops the uncertainty about the color of those three edges, the second replacement drops the conditions that those edges are not all the same color, and the `Append` asserts that indeed they are all red. The result `False` means that it is impossible to deny that there is a second monochromatic triangle, or, more simply, there is a second monochromatic triangle.

The Ramsey number, , is defined to be the least integer such that any red-blue edge coloring of results in either a blue or a red . (The previous section showed that .) Not many Ramsey numbers are precisely known [1, 2].

This confirms the result .

This confirms the result .

The calculation took too long and had to be aborted, but the simple idea to speed up the party problem in the previous section makes it possible to show : in any , at least half the edges from vertex 1 must be the same color; that is, edges must be the same color. Unless there is symmetry (i.e., ), we must consider two separate cases, that this color is blue or red. The function `QuickerRamseyTest` is the faster version of `RamseyTest`.

For the symmetric case , only one test suffices for each of and .

There are more than 6000 clauses in the cnf constructed to show that , since there are two clauses needed in to rule out any ’s being all red or all blue and there are 3060 in (). Also, each of these clauses has six negated propositional letters, one for each edge of the (a has edges). In addition, there are the clauses that require each edge of to be red or blue but not both, has edges, and so on. It seems that *Mathematica*’s claim of “industrial strength” Boolean capability is fully justified.

Ramsey theory also considers edge colorings with more than two colors. The classical result for three colors is ; that is, it is not possible to color the edges of with three colors without a monochromatic triangle; however, has such a coloring.

This generalizes `ColorEdges` from two to three colors so that each edge gets exactly one of three colors.

The generalization of `RamseyTest` takes an additional argument, , and a third color. It tests whether the edges of can be colored with three colors without a monochromatic , , or .

To speed up the calculation, we make use of the following observation. Vertex 1 of is connected by 16 edges to the remaining vertices. Surely, at least six of these edges must get the same color, say red; for if at most five got the same color, there would be at most 15 edges. This leads to a quicker test formula than `RamseyTest`.

To check , it suffices to check and .

Here is the with its edges colored red, blue, and green separately and together. There are no monochromatic triangles. Each of the three graphs with one color is known to be isomorphic to the Clebsch graph; [5] shows them with their vertices permuted in all possible ways.

Suppose we remove an edge from ; will it still be the case that any two-coloring of the edges has a monochromatic triangle? Or, if we remove an edge from , will the resulting graph still have the property that any two-coloring of its edges must yield a monochromatic ? We show that Boolean computation is also well suited to investigate these kinds of problems.

Remove from the clauses that require the edge to be blue or green, but not both, and add a statement to color that edge white. The result is satisfiable, which means that removing just one edge from gives a graph that can be two-colored without monochromatic triangles.

This defines an auxiliary function.

Here is the coloring.

Recall that ; however, removing even one edge from allows a two-coloring without a monochromatic .

Unlike the case of with an edge removed, it is too hard to see that the blue and green edge-colored subgraphs contain no monochromatic . `FindClique` finds the largest complete subgraph, which in both cases is a triangle, not a .

*Mathematica*’s Boolean computation is a useful tool for doing research in mathematics.

We thank the editor and referee for their generous help, which greatly improved this work.

[1] | E. W. Weisstein. “Ramsey Number” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/RamseyNumber.html. |

[2] | R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2nd ed., New York: Wiley-Interscience, 1990. |

[3] | A. W. Goodman, “On Sets of Acquaintances and Strangers at Any Party,” American Mathematical Monthly, 66(9), 1959 pp. 778-783. www.jstor.org/stable/2310464. |

[4] | G. Beck. “Among Six People, Either Three Know Each Other or Three Are Strangers to Each Other” from the Wolfram Demonstrations Project—A Wolfram Web Resource. www.demonstrations.wolfram.com/AmongSixPeopleEitherThreeKnowEachOtherOrThreeAreStrangersToE. |

[5] | E. Pegg Jr. “Ramsey(3,3,3)>16″ from the Wolfram Demonstrations Project—A Wolfram Web Resource. www.demonstrations.wolfram.com/Ramsey33316. |

R. Cowen, “Using Boolean Computation to Solve Some Problems from Ramsey Theory,” The Mathematica Journal, 2013. dx.doi.org/doi:10.3888/tmj.15-10. |

Robert Cowen is a professor of mathematics emeritus at Queens College, CUNY. He uses *Mathematica* in his own research and has written a textbook with John Kennedy called *Discovering Mathematics with Mathematica*. His web page can be found at sites.google.com/site/robertcowen.

**Robert Cowen**

*Department of Mathematics
Queens College, CUNY
Flushing, NY 11367*