# Magic "Squares" Indeed!

by Arthur T . Benjamin and Kan Yasuda

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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