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.
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).
So far, that’s as close as anyone has gotten.Will you be the first to solve either problem? Post solutions and magic squares of your own design below. If enough interesting submissions come in, some will be featured in a future blog post at www.ScientificAmerican.com
*Clarification (10/20/14): This sentence was edited after posting. The original did not specify that the numbers in each of the magic squares are different from one another.