Magic Squares of Squares

June 24, 2005 at 1:00 pm

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):