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

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## References

Borkovitz, D. R. and Hwang, F. K. "Multiplicative Magic Squares." Disc. Math. 47, 1-11, 1983.Boyer, C. "The Smallest Possible Multiplicative Magic Squares." http://www.multimagie.com/English/Multiplicative.htm.Dudeney, 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. http://www.maa.org/editorial/mathgames/mathgames_11_07_05.html.Sloane, N. J. A. Sequence A114060 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Multiplication Magic Square

## Cite this as:

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