MULTIMAGIE.COM - 4x4 to 7x7 magic squares of squares

First known 4x4 to 7x7 magic squares of squares
See also the Magic squares of squares page

Can a 3x3 magic square be constructed with nine distinct square numbers? The answer is unknown today: nobody has succeeded in constructing a 3x3 magic square of squares, and nobody has proved that it is impossible to construct such a square. See my article in The Mathematical Intelligencer, and the Powerpoint file of the lecture.

But it is possible to construct other magic squares of squares:

Well-known 8x8 (and greater) bimagic squares are of course magic squares of squares because, by definition, they remain magic when their numbers are squared. And, as required, all their numbers are distinct because they use consecutive integers. Unfortunately, it is impossible to construct 7x7 (and smaller) bimagic squares using consecutive integers.
Between the unknown 3x3 and the well-known bimagic 8x8 (and greater), magic squares of squares from 4x4 to 7x7 are known today as I will show in this page.

On this subject, read also the MathTrek article written in June 2005 by Ivars Peterson:

4x4 magic square of squares

The first known magic square of squares was sent in 1770 by Leonhard Euler to Joseph Lagrange. This is the square LE2 fully explained and described in the M.I. article (and lecture slides 22 and 23).

1770: 4x4 magic square of squares, by Leonhard Euler.
S2 = 8515
68²	29²	41²	37²
17²	31²	79²	32²
59²	28²	23²	61²
11²	77²	8²	49²

This square can be constructed with his wonderful 4x4 parametric solution and (a, b, c, d, p, q, r, s) = (5, 5, 9, 0, 6, 4, 2, -3). Or also (5, 5, 9, 0, 2, 3, 6, –4), giving the same square but permuted. From my M.I. paper published in 2005:

"The work of Euler is linked to the theory of quaternions [2] [15] [36] [37] developed later in 1843 by William Hamilton. In his (LE3) square, Euler reused an identity that he found and sent to Christian Goldbach in 1748 [21]:

(a² + b² + c² + d²)(p² + q² + r² + s²) = (ap + bq + cr + ds)² + (aq - bp - cs + dr)² + (ar + bs - cp - dq)² + (as - br + cq - dp)²

This identity also follows from the fact that the norm of the product of two quaternions is the product of the norms. Euler first used this identity in 1754 [17] in a partial proof that every positive integer is the sum or at most four square integers, an old conjecture announced by Diophantus, Bachet, and Fermat. Using as a basis these partial results of Euler's, Lagrange published in 1770 [55] the first complete proof of this four square theorem, the same year as the letter he received with the first 4x4 magic square of squares".

In this article and its supplement of 2005, I gave two sub-families (CB2) and (CB15) of Euler's parametric solution, but using only one parameter k instead of the eight parameters (a, b, c, d, p, q, r, s) from Euler. With k = 5, my sub-family (CB15) produces the above Euler-Lagrange example.

In November 2018, Seiji Tomita, Japan, found 11 more parametric solutions also using only one parameter k, an excellent set of solutions. As it was with my two parametric solutions, for each k, we also need to check that the 16 produced cells are distinct. And as it was with my two parametric solutions, I remark that his 11 solutions are also sub-families of Euler's parametric square (a, b, c, d, p, q, r, s), setting b = k and d = 0:

(9, k, 16, 0, 1, 2, 6, -3)
(5, k, 16, 0, 2, 3, 9, -6)
(7, k, 18, 0, 3, 4, 8, -6)
(5/2, k, 9/2, 0, 1, 3, 15, -5)
(5, k, 7, 0, 2, 4, 10, -5)
(2, k, 5, 0, 3, 5, 15, -9)
(5/2, k, 7/2, 0, 3, 7, 21, -9)
(5, k, 16, 0, 1, 2, 10, -5)
(2, k, 9, 0, 2, 3, 12, -8)
(1, k, 7, 0, 2, 3, 18, -12)
(7, k, 10, 0, 6, 8, 12, -9)

5x5 magic square of squares

In 2004, I constructed the first known 5x5 magic squares of squares. Squares CB4 and CB5 published in the M.I. article (and lecture slide 17). The smallest possible is CB4:

2004: 5x5 magic square of squares.
S2 = 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

6x6 magic square of squares

Unfortunately to late to be published in the M.I. article, I constructed, in June 2005, the first 6x6 magic squares of squares.

If I am right, 6x6 magic squares of squares using squared consecutive integers (0² to 35², or 1² to 36²) are impossible. My 6x6 magic square of squares does NOT use squared consecutive integers... but it is interesting to see the used numbers:

from 0² to 36² only excluding 30².

It is impossible to construct a 6x6 magic square of squares with a smaller magic sum. But it is possible to construct other samples with the same magic sum S2 = 2551, or with other bigger sums.

*2005: 6x6 magic square of squares. S2 = 2551.*
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

An interesting supplemental characteristics of this sample: the 3 smallest integers (0², 1², 2²) and the 2 biggest (35², 36²) are used together in the first row.

7x7 magic square of squares

Unfortunately to late to be published in the M.I. article, I constructed, in June 2005, the first 7x7 magic squares of squares.

The smallest order allowing magic squares of squares using squared consecutive integers is the order 7. An indirect consequence: the impossibility of 7x7 bimagic squares is not coming from a problem with its squared numbers!

Here is my sample using the squared integers from 0² to 48²:

*2005: 7x7 magic square of squares. S2 = 5432*
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

An interesting supplemental characteristics added in this sample: the 7 rows are magic (S1=168) when the integers are not squared, meaning that the 7 rows are bimagic!

Conclusion of this page: because 4x4 and above are now solved, it means that 3x3 is the only remaining open problem (but the most difficult...) on magic squares of squares!

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