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Math Trek

Magic Squares of Squares

1:00pm, June 24, 2005

People have been toying with magic squares for more than 2,000 years—setting themselves increasingly difficult challenges to find arrays of numbers that fit given patterns.

Typically, a magic square consists of a set of distinct integers arranged in the form of a square so that the numbers in each row, column, and diagonal all add up to the same total. The most recent developments concern magic squares in which each of the entries is a different squared number—a magic square of squares.

In the current Mathematical Intelligencer, Christian Boyer of Enghien les Bains, France, summarizes progress in finding magic squares of squares and lists several unsolved problems involving such patterns. In one case, he offers a cash (and champagne) prize for the solution.

The first known magic square of squares was devised by Leonhard Euler (1707–1783). He described this four-by-four array in a letter he sent in 1770 to Joseph-Louis Lagrange (1736&ndas

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