## Constructing Crossword Arrays Faster »

### T R O T T ' S C O R N E R

We implement an algorithm to build a crossword array for a set of given words based on hashing techniques. Read More »

Published November 23, 2009

We implement an algorithm to build a crossword array for a set of given words based on hashing techniques. Read More »

Published November 23, 2009

Sudoku is just one of hundreds of great puzzle types. This column presents obscure logic puzzles of various sorts and challenges the readers to solve the puzzles in two ways: by hand and with *Mathematica*. For the latter, solvers are invited to send their code to edp@wolfram.com. The person submitting the most elegant solution will receive a prize. Read More »

Published November 23, 2009

The eighth International *Mathematica* Symposium (*IMS*’06) was held in France for the first time. This initiative to promote scientific research and exchange in the broad community of enthusiastic users was held in June 2006 at the “Palais des Papes” of Avignon, next to the celebrated “pont St Benezet”—two monuments inscribed by UNESCO on the World Heritage List with the “Rocher des Doms”. Read More »

Published November 23, 2009

Computing accurate approximations to the perimeter of an ellipse is a favorite problem of mathematicians, attracting luminaries such as Ramanujan [1, 2, 3]. As is well known, the perimeter of an ellipse with semimajor axis and semiminor axis can be expressed exactly as a complete elliptic integral of the second kind. Read More »

Published November 23, 2009

An algorithmic approach to manifolds is presented, based on an object approach to the parametric plotting commands. The initial purpose was to blend geometric and symbolic aspects, so as to equip computer-assisted design (CAD) with symbolic capabilities. Nevertheless, this investigation aims more generally at providing a uniform treatment of analytic geometry and field analysis, in view of applications to physics, system modeling, and morphology. Read More »

Published November 23, 2009

Visualization is an invaluable companion to symbolic computation in understanding the complex plane and complex-valued functions of a complex variable. The *Presentations* application, an add-on to *Mathematica,* provides a rich set of tools for assisting such visualization. This article demonstrates some capabilities of the application, especially those relevant in an introduction to complex analysis, and it indicates some teaching and learning issues that arise. Included are examples of how complex functions map objects in the complex plane and on the Riemann sphere, and of how complex functions behave near singularities and at branch points. Read More »

Published November 23, 2009

In parameterized mechanism design, there are two contradicting requirements: keep the governing equation as general as possible (everything in symbolic form), but be able to quickly look at (or simulate) the mechanism and see how it moves at every stage of the design process (which requires numeric representation). Handling symbolic and numerical representations together is a basic paradigm of symbolic manipulator programs like *Mathematica*. The solutions developed for this paradigm can be used for enhancing the flexibility of a mechanism prototyping application package written in *Mathematica*. This article shows some examples of mechanism modeling with the *LinkageDesigner* application package that uses this bundled symbolic and numeric representation. The bridge between the two representations is substitution, enabling a one-way route from symbolic to numeric representation. A historical parabola-drawing mechanism and the spirograph will be used as examples. Besides the power of substitution, replacement is also widely used in *LinkageDesigner*. This simple tool can be handy in solving problems like the inverse kinematic problem (IKP). Two examples will be considered: the IKP of a 6R robot and a parallel mechanism with three degrees of freedom. Read More »

Published November 23, 2009

We discuss a new tool that is successfully used in an ongoing project to compute : an automated symbolic induction prover (SIP). The SIP tool is written in *Mathematica* and provides a unified way to prove that large sets of Turing machines are nonhalters. In a way, an SIP enables certain Turing machines to provide their own proof of being a nonhalter. Read More »

Published November 23, 2009

Imagine a floor marked with many equally spaced parallel lines and a thin stick whose length exactly equals the distance between the lines. If we throw the stick on the floor, the stick may or may not cross one of the lines. The probability for a hit involves . This is surprising since there are no circles involved; on the contrary, there are only straight lines. If we repeat the experiment many times and keep track of the hits, we can get an estimate of the irrational number . (We also consider sticks of length .) Read More »