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
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).