What is a multiplicative magic square?
It is a square which is magic using multiplication instead of addition. A “multiplicative” magic square is very easy to construct from a standard “additive” magic square, using the numbers of the additive magic square as powers of a fixed integer. For example: Construction of a multiplicative magic square P=4096, from a standard additive magic square S=12. Additive =12 >> (powering step) >> Multiplicative =4096 3 8 1 =12 23 28 21 8 256 2 =4096 2 4 6 =12 22 24 26 4 16 64 =4096 7 0 5 =12 27 20 25 128 1 32 =4096 =12 =12 =12 =12 =4096 =4096 =4096 =4096 When we add numbers of any line in the left square, we get always the same number (here 12). When we multiply numbers of any line in the right square, we get always the same number (here 4096). This 3rd-order multiplicative square on the right was published by Antoine Arnauld in Nouveaux Elments de Gomtrie, Paris, in… 1667… a long time ago! Some properties on multiplicative magic squares: • It is impossible to construct a multiplicative
