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.

### Instructions

During the 1890s, puzzler Sam Loyd wrote columns for the *Brooklyn Daily Eagle*. On February 28, 1897, the following puzzle appeared [1].

(The solution to this puzzle is on page 6 of the online notebook. The online notebook contains code for the Puzzleland Park layout and code for the layout of the solution.)

This puzzle type is now a regular feature of Japanese puzzle magazines under the name Number Link. The format has been refined in the last hundred years. Each puzzle follows these rules.

- Connect identical numbers with a continuous path.
- Paths must go through the center of a cell horizontally or vertically and never go through the same cell twice.
- Paths cannot cross, branch off, or go through other numbered cells.
- Every unnumbered square must contain part of a path.

### Example

Here is an example of a Number Link puzzle and its solution.

Finding the solution can start with the 1 in the corner—the path must go up, then right, then up. That forces the path of the 2 above it to start going up and the path of the 4 on the right to start going right.

### Puzzles

Here are four sample puzzles from Penpa Mix #2 [2]. Each has a unique solution, which can be found by hand. In the first puzzle, each of the numbers in the corners has a forced starting path. What techniques are necessary to complete a solution by hand? A more interesting question is how these can be solved programmatically.

### Puzzle Source

### Previous Issue’s Solution

The 10:2 column discussed Ripple Effect puzzles. Yves Papegay sent a complete solution, which is available in ripple.nb. His solution qualifies him for *The Mathematica Guidebook* of his choice.

Yves’ solution has two parts. First, he checks each cell of each room for the main criteria: only once, not too close, and at least once. These three criteria form what he calls a *Naive* filter. Surprisingly, multiple uses of the *Naive* filter will solve most of the Ripple Effect puzzles given in the previous column. For the remainder, he introduces code to fix all possible values in each cell with a *One Step Further* filter. Fixed values that then fail to give solutions with the *Naive* filter are then discarded as impossible.

### References

[1] | S. Loyd, Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums (With Answers), New York: The Lamb Publishing Company, 1914, p. 61.www.mathpuzzle.com/loyd/cop060-061.html. |

[2] | Penpa Mix #2, Tokyo, Japan: NIKOLI Co., Ltd., 2005, pp. 36-37.www.nikoli.co.jp/howtoget-e.htm. |

E. Pegg Jr, “Number Link,” The Mathematica Journal, 2012. dx.doi.org/10.3888/tmj.10.3-2. |

### Additional Material

ripple.nb

Available at content.wolfram.com/sites/19/2007/08/ripple.nb.

### About the Author

**Ed Pegg Jr**

Scientific Information Editor

*Wolfram Research, Inc.
*

*edp@wolfram.com*