Group table and Sudoku puzzles




Wu, Qingquan

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For any finite group, we will notice a striking similarity between its group multiplication table and the Sudoku puzzles. Every nXn Sudoku puzzle should satisfy three rules: Every row should contain exactly those n numbers 1 through n; Every column should contain exactly those n numbers 1 throught n; In addition, if n=kXk is a perfect square, then every kXk (non-overlapping) grid should contain exactly those n numbers 1 through n. By the cancellation law of the group, every group multiplication table will automatically satisfy the first two rules. Unfortunately, it will almost always fail the last rule. One way to fix it is to allow row/column switching for the group multiplication table. A natural question is: Can all Sudoku puzzles be induced by a group in this way? The answer is: It depends. We will explore this question from both algebraic and statistical perspectives and search through computer programming to see the percentage of group-induced Sudokus among all Sudokus.


When I was teaching algebraic structure to the engineering students, I noticed the difficulty of introducing abstract structures. Group multiplication table will convey all of the information of a group, but it is more visual-friendly. So I switched to the group tables. It is very noticeable that the table looks like a (complete) Sudoku puzzle. So a natural question is: Will the group table always become a Sudoku puzzle? This is the origin of the research project. Then I assigned it to two undergraduates to become a research project. We finished most of the theoretical work and we are currently in searching for a good Sukodu-generating algorithm so that we can verify how many of those Sudokus are induced by groups.


2023 Faculty and Student Research Poster Session and Research Fair, West Texas A&M University, College of Engineering, Poster, Sudoku puzzles, Multiplication table, Mathematics


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