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# Magic Squares and Sudoku

John Lorch
The American Mathematical Monthly
Vol. 119, No. 9 (November 2012), pp. 759-770
DOI: 10.4169/amer.math.monthly.119.09.759
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.119.09.759
Page Count: 12
Item Type
Article
References
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## Abstract

Abstract We introduce a family of magic squares, called linear magic squares, and show that any parallel linear sudoku solution of sufficiently large order can be relabeled so that all of its subsquares are linear magic. As a consequence, we show that if n has prime factorization \documentclass{article} \pagestyle{empty}\begin{document} $p_1^{k_1}\cdots p_t^{k_t}$ \end{document} and \documentclass{article} \pagestyle{empty}\begin{document} $q=\min \{p_j^{k_j}\mid 1\leq j\leq t\}$ \end{document} , then there is a family of q(q − 1) mutually orthogonal magic sudoku solutions of order n2 whenever q > 3; such an orthogonal family is complete if n is a prime power.