Implications of Sudoku Boards as Matrices
Location
Access Type
Campus Access Only
Entry Number
35
Start Date
4-7-2021 4:15 PM
End Date
4-7-2021 4:30 PM
Department
Mathematics
Abstract
This project focuses on a variant of the popular game known as Sudoku puzzles. Sudoku boards consist of a 9 x 9 grid composed of nine 3 x 3 boxes that contain nine individual squares each. A valid Sudoku board consists of the integers 1 through 9 placed exactly once in each box, column, and row in the board. This study focuses on the 4 x 4 variants of Sudoku boards called Shidoku boards. The researcher treated Shidoku boards as matrices to examine patterns that produced determinants of zero. Determinants are a numerical value that can be obtained from all square n x n matrices. These values are of interest to researchers as they influence certain properties of square matrices such as if a matrix has an inverse. This study found that a Shidoku board with two repeated pairs of integers on its main diagonal is guaranteed to have a determinant of zero regardless of integer placement in the board. Additionally, the researcher conducted an exhaustive search of Shidoku board determinants and found that 160 out of all 288 possible Shidoku boards have determinants of zero. The researcher found that all 12 potential patterns for Shidoku boards produce at least eight determinants of zero under the permutations of 1, 2, 3, and 4. Furthermore, the researcher also examined the determinants for the sums and differences of pairs of the 12 potential patterns for Shidoku boards.
Faculty Mentor(s)
Dr. Danny ClineDr. Jennifer StyrskyDr. Leslie Hatfield
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Implications of Sudoku Boards as Matrices
This project focuses on a variant of the popular game known as Sudoku puzzles. Sudoku boards consist of a 9 x 9 grid composed of nine 3 x 3 boxes that contain nine individual squares each. A valid Sudoku board consists of the integers 1 through 9 placed exactly once in each box, column, and row in the board. This study focuses on the 4 x 4 variants of Sudoku boards called Shidoku boards. The researcher treated Shidoku boards as matrices to examine patterns that produced determinants of zero. Determinants are a numerical value that can be obtained from all square n x n matrices. These values are of interest to researchers as they influence certain properties of square matrices such as if a matrix has an inverse. This study found that a Shidoku board with two repeated pairs of integers on its main diagonal is guaranteed to have a determinant of zero regardless of integer placement in the board. Additionally, the researcher conducted an exhaustive search of Shidoku board determinants and found that 160 out of all 288 possible Shidoku boards have determinants of zero. The researcher found that all 12 potential patterns for Shidoku boards produce at least eight determinants of zero under the permutations of 1, 2, 3, and 4. Furthermore, the researcher also examined the determinants for the sums and differences of pairs of the 12 potential patterns for Shidoku boards.