Multiplications—for a change! | Tangente

The first multiplicative magic square seems to have been published by the theologian and mathematician Antoine Arnauld (1612–1694) in his Nouveaux Éléments de géométrie (1683).

This object was constructed from the 3 × 3 additive square (the Lo-shu, known in China for more than twenty-two centuries). Because exponents add when powers of the same number are multiplied, the magic-sum property is preserved when the entries of the additive square are replaced by powers of a single integer, with the entries of the original square as exponents. This yields a multiplicative square with a magic product of 2¹⁵, or 32,768. But by reducing every exponent by 1 (equivalent to starting with an additive square containing the numbers from 0 to 8 rather than the integers from 1 to 9), Arnauld obtained a magic product of 2¹², or 4,096.