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Magic Squares of Squares
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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 (1736–1813).

 682 292 412 372 172 312 792 322 592 282 232 612 112 772 82 492

In Euler's magic square of squares (above), the four rows, four columns, and two diagonals each have the sum 8515.

In a separate publication, Euler revealed the formula that he had used to come up with this pattern—one member of a family of magic squares of squares.

 (+ap+bq+cr+ds)² (+ar–bs–cp+dq)² (–as–br+cq+dp)² (+aq–bp+cs–dr)² (–aq+bp+cs–dr)² (+as+br+cq+dp)² (+ar–bs+cp–dq)² (+ap+bq–cr–ds)² (+ar+bs–cp–dq)² (–ap+bq–cr+ds)² (+aq+bp+cs+dr)² (+as–br–cq+dp)² (–as+br–cq+dp)² (–aq–bp+cs+dr)² (–ap+bq+cr–ds)² (+ar+bs+cp+dq)²

The magic sum is given by (a² + b² + c²

+ d²)(p² + q² + r² +

s²).

Two additional conditions are needed to get the diagonals to add up to the same sum:

• pr + qs = 0,
• a/c = [–d(pq + rs) –

b(ps + qr)]/[b(pq + rs)

+ d(ps + qr)].

To get the magic square that he sent to Lagrange, Euler set a = 5,

b = 5, c = 9, d = 0, p = 6, q =

4, r = 2, s = –3.

The smallest magic square of squares belonging to this family (below) is not one found

by Euler himself. It's generated by setting a = 2, b = 3,

c = 5, d = 0, p = 1, q = 2, r =

8, s = –4. The magic sum is 3230.

 482 232 62 192 212 262 332 322 12 362 132 422 222 272 442 92

There are other four-by-four patterns now known that do not belong to Euler's formula family.

Boyer has recently come up with the first examples of five-by-five, six-by-six, and seven-by-seven magic squares of squares.

Here's the smallest five-by-five magic square of squares, with a magic sum of 1375.

 12 22 312 32 202 222 162 132 52 212 112 232 102 242 72 122 152 92 272 142 252 192 82 62 172

Boyer suspects that it's impossible to get a six-by-six magic square using squared consecutive integers (02 to 352 or 12 to 362). Interestingly, the example that he did find (below) has all the squares from 02 to 362 with the exception of 302.

 22 12 362 52 02 352 62 332 202 292 42 132 252 72 142 242 312 122 212 322 112 152 222 162 342 182 232 102 192 92 172 82 32 282 272 262

Boyer's seven-by-seven magic square of squares with the smallest magic sum (below), however, does use squared consecutive integers.

 252 452 152 142 442 52 202 162 102 222 62 462 262 422 482 92 182 412 272 132 122 342 372 312 332 02 292 42 192 72 352 302 12 362 402 212 322 22 392 232 432 82 172 282 472 32 112 242 382

Is there a three-by-three magic square of squares? No one knows.

In 1996, Martin Gardner offered \$100 as a prize to the first person to construct such a square from nine distinct integer squares. No one has yet come forward with an example—or a proof that it's impossible to create one.

Boyer has offered a prize of €100 (and a bottle of champagne) for a solution to a presumably easier problem: finding a new example of a three-by-three magic square with seven squared entries that differs from the one already known or constructing the first example with eight squared entries.

Here's the known example of a three-by-three magic square in which seven of the entries are squared integers, found by Andrew Bremner of Arizona State University (and independently by Lee Sallows of the University of Nijmegen):

 3732 2892 5652 360721 4252 232 2052 5272 222121

Of course, rotations, symmetries, and multiples of this known square don't count as new solutions.

What is known about any three-by-three magic square of nine squared integers—if it exists—is that the numbers involved would be huge.

Happy hunting!

Comment

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• Dear Ivars,
...
there's a subtle reason for "why a 3x3-normal magic square with 'square' entries doesn't exist." if we examined the 3x3-normal magic square, we would immediately notice this more basic relationship:
...
A B C
...
D E F
...
G H I
...
for n = 3 (ONLY), we see that B +H = 2*E, D +F = 2*E, G +C = 2*E, and A +I = 2*E. so, if we let each entry be a 'square', then by substitution, we have...
...
a^2 b^2 c^2
...
d^2 e^2 f^2
...
g^2 h^2 i^2
...
thus, if we chose any equation, then we'd have x^2= 2*z^2 -y^2. however, with Pythagorean triples from... a^2 = b^2 -c^2, solutions ONLY exist when a, b, and c are co-prime, and ONLY when b^2 -c^2 can be factored properly. but, this isn't the case with 2*z^2 -y^2; it can't be factored. please verify using WolframAlpha that x, y, z = 0 when you plug in x^2 +y^2 = 2*z^2. thus, a 3x3-normal magic square w/square entries can't exist! (and some people thought that the solution to this easily-defined problem would be larger numbers than a modern-day computer could hold.)
*QED
1/17/2013
...
I should be able to collect any/all prize monies. Also, you MUST !!omit!! Chrisitian Boyer's 6x6 & 7x7 MSQ's of SQs. An entry cannot be 0^2 and be considered a "normal" magic sq.
...
Thanks, Bill
Bill Bouris
Mar. 18, 2013 at 10:14am