The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
Integer Programming Model for the Sudoku Problem, Bartlett, Chartier, Langville, Rankin

## 5. Conclusion

This paper examined the popular Sudoku puzzles from two angles: puzzle solution and puzzle creation. The first portion of the paper presented a binary integer programming formulation that solves any n \times n Sudoku puzzle. A Matlab m-file, which executes a branch and bound solution method, is available for download. Further, such an approach was extended to variations on the traditional Sudoku puzzle. The second half of the paper presented theorems for creating new Sudoku puzzles. We discovered that, starting with one Sudoku puzzle, we can easily produce a daily calendar of Sudoku puzzles (enough for the entire next century!). By adding or removing givens, we can also vary the level of difficulty of the games. Answers to the exercises are provided in the following pages, and we hope students attempt and enjoy the open-ended challenge questions.

### Acknowledgements

The authors thank the National Science Foundation as a portion of this work was supported under NSF grant CAREER-0546622.

Gratitude is also expressed to Philip Riley and Laura Taalman for creating the puzzles in this paper. For more puzzles see their book Color Sudoku or visit Brainfreeze Puzzles.

The Java applications in this article were created using Easy Java Simulations. For more information, visit http://www.um.es/fem/Ejs. The authors thank Wolfgang Christian of Davidson College and Francisco Esquembre of the University of Murcia for their guidance in using the Easy Java Simulations software for this article.