Can You Solve A Puzzle Unsolved Since 1996?‏

For thousands of years people have played with magic squares—arrays of distinct numbers whose rows, columns and diagonals add to the same total.* A simple 3-by-3 array that sums to 15 every which way appears on the back of a turtle in the legend of Lo Shu, a Chinese tale from 650 B.C. Medieval mathematicians in the Middle East and India studied magic squares of varying sizes and Albrecht Dürer included a 4-by-4 magic square in his famous engraving, Melencolia I, in 1514. Today amateur and professional mathematicians continue to devise new magic squares, even adding extra dimensions to envision 3-D magic cubes and 4-D magic tesseracts.

Leonhard Euler, an 18th-century mathematician, puzzled over another type of exotic magic square, one made entirely of squared numbers. In 1770 he introduced the first 4-by-4 example of a magic square of squares (below), along with a formula for producing others.

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Many 4-by-4 magic squares of squares are now known and about 10 years ago Christian Boyer reported the first examples of 5-by-5, 6-by-6 and 7-by-7 magic squares of squares. To date, though, no one has discovered a 3-by-3 magic square of squares nor has anyone proved it impossible.
In 1996 Martin Gardner, who had written Scientific American’s Mathematical Games column for some 25 years, offered a $100 prize to anyone who could devise a solution. A year later magic square expert Lee Sallows described a near miss (see below), with only one diagonal summing differently (going from the top left to the bottom right yields 38,307, not 21,609—the total in all other directions).

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So far, that’s as close as anyone has gotten.Will you be the first to solve either problem?



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