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Elizabeth Arnold

#### Abstract

Recently there has been a lot of interest in the mathematics of the popular game Sudoku. In a typical Sudoku puzzle, a number of initial clues are given, and the solver uses strategies to fill in the remaining clues to complete the board. A well-known open problem is, “How many initial clues are necessary for the puzzle to have a unique completion?” In this talk, we shift the focus of study from clues to what we call packets. A packet gives information about what entries can NOT be in a cell. Introducing packets gives rise to some interesting questions about Sudoku and its $4\times4$ counterpart, Shidoku. One such question is what is the minimum number of packets needed to describe a puzzle with a unique completion?'' This question parallels the minimum clue question. Packets are also intimately related to the Boolean system of polynomial equations used to describe the constraints of a Sudoku puzzle. They can be used to more efficiently calculate a Gr\"obner basis of the ideal generated by this system of equations. Packets are also inherently related to human methods for solving Sudoku puzzles. To emulate human solving strategies we introduce the idea of solving symmetries -- functions which manipulate a puzzle while maintaining the same solutions. We show that these solving symmetries form a group which acts on the set of Sudoku puzzles.

#### Cover Page Note

This research was performed at a summer REU at James Madison University and supported by NSF DMS-1004516. Foremost we want to thank our advisor Dr. Elizabeth Arnold for her continued dedication in helping us on this project. We would also like to thank our REU colleagues Bjorn Wastvedt, Eddie Tu, and Dr. Stephen Lucas for their help and support.

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