Multiplication Magic Square

A square which is magic under multiplication instead of addition (the operation used to define a conventional magic square) is called a multiplication magic square. Unlike (normal) magic squares, the n^2 entries for an nth order multiplicative magic square are not required to be consecutive.


The above multiplication magic square has a multiplicative magic constant of 4096 and was found by Antoine Arnauld in Nouveaux Eléments de Géométrie, Paris in 1667 (Boyer).


The smallest possible magic constants for 3×3, 4×4, ... are 216, 5040, 302400, 25945920, ... (OEIS A114060). The 3×3 solution (left) was found by Sayles in 1913 and also published by Dudeney (1917). Sayles also found the 4×4 solution (right), which was subsequently proved to be minimal by Borkovitz and Hwang (1983). The series of best known smallest largest element for an n×n multiplication magic square with n=3, 4, ... begins 36, 28, 45, 66, 91, 160, 225, ... (Boyer).

See also

Addition-Multiplication Magic Square, Magic Square

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Borkovitz, D. R. and Hwang, F. K. "Multiplicative Magic Squares." Disc. Math. 47, 1-11, 1983.Boyer, C. "The Smallest Possible Multiplicative Magic Squares.", H. E. "Chessboard Problems." Amusements in Mathematics. 1917. Reprinted as New York: Dover, 1970.Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 30-31, 1975.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 89-91, 1979.Pegg, E. "Math Games: Times Square Magic." Nov. 14, 2005., N. J. A. Sequence A114060 in "The On-Line Encyclopedia of Integer Sequences."

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Multiplication Magic Square

Cite this as:

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

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