An addition-multiplication square is a square of integers that is simultaneously a magic square and multiplication magic square.
In 1955, Horner found a square of order eight with addition magic constant 840 and multiplicative magic constant 2058068231856000, and largest number 261 (left figure; Horner 1955, Hunter and Madachy 1975). In 2005, Boyer found a smaller order-eight square above with addition constant 600, multiplicative constant 67463283888000, and largest number 225 (right figure). Boyer also found a square with smaller multiplicative constant 51407948592000, but larger addition constant (760) and largest number (333).
The two addition-multiplication squares above have order nine with addition magic constants 848 and 1200 and multiplicative magic constants 
and 
, respectively (Hunter and Madachy 1975,
Madachy 1979).
L. Sallows has constructed an interesting 
magic square in which the rows and columns have constant
sum 0, while the toroidal diagonals have constant product 72.
