Bimagic Square

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.

Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times.


The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and 11180 after squaring. Another order 8 bimagic square is shown at right.

Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).


Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as puzzles to the reader and their solutions appearing two weeks later in the following issues (Boyer).


Wroblewski found the first known 6×6 bimagic square using distinct (but nonconsecutive) integers (Boyer 2006), illustrated above.

See also

Bimagic Cube, Magic Square, Multimagic Square, Panmagic Square, Trimagic Square

Explore with Wolfram|Alpha


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 212, 1987.Benson, W. H. and Jacoby, O. New Recreations with Magic Squares. New York: Dover, 1976.Boyer, C. "Multimagic Squares.", C. "Bimagic Squares.", C. "Smallest Bimagic Square.", C. "Multimagie News." Apr. 4, 2006., J. R. "Note on the Bimagic Square of Order 3." J. Recr. Math. 29, 265-267, 1998.Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, p. 31, 1975.Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical Recreations. New York: W. W. Norton, pp. 143 and 176-178, 1942.Lucas, E. "Les carrés magiques. Sur le carré de 3 et sur les carrés à deux degrés." Les Tablettes du Chercheur. No. 5, p. 7, March 1, 1891.Pfeffermann, G. "Carré magique à deux degrés." Les Tablettes du Chercheur. No. 2, p. 6, January 15, 1891a.Pfeffermann, G. "Carré magique de 9 à deux degrés." Les Tablettes du Chercheur. No. 14, pp. 5-6, July 15, 1891b.

Referenced on Wolfram|Alpha

Bimagic Square

Cite this as:

Weisstein, Eric W. "Bimagic Square." From MathWorld--A Wolfram Web Resource.

Subject classifications