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# 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 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 bimagic square using distinct (but nonconsecutive) integers (Boyer 2006), illustrated above.

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

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## References

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." http://www.multimagie.com/indexengl.htm.Boyer, C. "Bimagic Squares." http://www.multimagie.com/English/Bimagic.htm.Boyer, C. "Smallest Bimagic Square." http://www.multimagie.com/English/Smallestbi.htm.Boyer, C. "Multimagie News." Apr. 4, 2006. http://www.multimagie.com/English/News0604.htm.Hendricks, 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.

Bimagic Square

## Cite this as:

Weisstein, Eric W. "Bimagic Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BimagicSquare.html