66 // Squares of Squares

Squares of Squares Magic squares are so well known that I'm not going to say a lot about the common ones, but some of the variants are more interesting. For instance, is it possible to make a magic square whose entries are all distinct perfect squares? Call this a square of squares. (Clearly the condition of using consecutive whole numbers must be ignored!)

We still have no idea whether a 363 square of squares exists. Near misses include Lee Sallows's

1272 462 52

2 2

2 113 942

742 822 972

for which all rows, columns, and one diagonal have the same sum. Another near miss is magic:

3732 2892 5652

360;721 4252 232

2052 5272 222;121

However, only seven entries are square ­ I've marked the exceptions in bold. It was found by Sallows and (independently) Andrew Bremner.

In 1770 Euler sent the first 464 square of squares to Joseph- Louis Lagrange:

682 292 412 372

172 312 792 322

592 282 232 612

2 2 2

11 77 8 492

It has magic constant, 8515.

Christian Boyer has found 565; 666 and 767 squares of

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