The Puzzling Mathematics of Sudoku
Sudoku is the latest craze in puzzles, and is played by entering the digits from 1 to 9 to complete a partially filled 9x9 grid so that each digit appears exactly once in each row, column and 3x3 subgrid. There are numerous game variations, each of which has additional restrictions. The focus of my research was to investigate a few of these variations, for example, Sudoku X, where the entries on each of the main diagonals must be distinct. Another variation, Rainbow Sudoku, has nine additional colored regions which wrap around the board in a torus fashion, each of which must also be distinct. I have developed various algorithms for creating Sudoku boards which satisfy the additional conditions, and found a minimal set of operations for each set of boards generated which form an equivalence relation.
There are roughly different Sudoku boards, a value which has only been calculated through the use of computers. While the total number Sudoku X boards remains an open question, I have been able to determine the number of distinct diagonalizations using topics from discrete mathematics, such as permutations with restricted position and rook polynomials. Future areas of research include determining whether each diagonalization can be extended to a completed board, and to how many completed boards each diagonalization may be extended.
Faculty Mentor: Shelly Smith, Mathematics
April presented at MathFest, the Annual Summer Meeting of the Mathematics Association of America, July 31-August 3, 2008.
Page last modified July 14, 2009