First known 4x4 to 7x7 magic squares of squares

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).

 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:

1. (9, k, 16, 0, 1, 2, 6, -3)
2. (5, k, 16, 0, 2, 3, 9, -6)
3. (7, k, 18, 0, 3, 4, 8, -6)
4. (5/2, k, 9/2, 0, 1, 3, 15, -5)
5. (5, k, 7, 0, 2, 4, 10, -5)
6. (2, k, 5, 0, 3, 5, 15, -9)
7. (5/2, k, 7/2, 0, 3, 7, 21, -9)
8. (5, k, 16, 0, 1, 2, 10, -5)
9. (2, k, 9, 0, 2, 3, 12, -8)
10. (1, k, 7, 0, 2, 3, 18, -12)
11. (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:

 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.

 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:

 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!