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 today unknown: nobody has succeeded to construct 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:

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).


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:


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 (0 to 35, or 1 to 36) are impossible. My 6x6 magic square of squares does NOT use squared consecutive integers... but it is interesting to see the used numbers:

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.

An interesting supplemental characteristics of this sample: the 3 smallest integers (0, 1, 2) and the 2 biggest (35, 36) 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 0 to 48:

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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