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Magic "Squares" Indeed!

by Arthur T . Benjamin and Kan Yasuda

This article originally appeared in:
American Mathematical Monthly
February, 1999

Subject classification(s): Algebra and Number Theory | Linear Algebra | Matrix Algebra | Number Theory | Discrete Mathematics | Game Theory
Applicable Course(s): 3.8 Linear/Matrix Algebra | 4.3 Number Theory

A real matrix is called square-palindromic if, for every base \(b\), the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when the numbers are read "backwards" in base \(b\).  The authors prove that all \(3 \times 3\) magic squares are square-palindromic.  They also give sufficient conditions for  \(n \times n\) magic squares to be square-palindromic, which include all circulant matrices and all symmetrical magic squares.

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Capsule Course Topic(s):
Linear Algebra | Matrix Algebra
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