Ignore the desire for entries to be distinct for a minute. There is a 4x4 magic square with sum 1: 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 It's unique, up to rotation and reflection. You can tile the 4x4 square with four of these: A B C D C D A B D C B A B A D C Looks suspiciously familiar from other 4x4 constructions, doesn't it? It's also unique, up to taking the transpose. Take ordered pairs of this with its transpose, and you get Aa Bc Cd Db Cb Dd Ac Ba Dc Ca Bb Ad Bd Ab Da Cc Hooray, these sixteen are all distinct. Under addition, you can let (a,b,c,d)=(0,1,2,3) and (A,B,C,D)=(00,10,20,30) base 4, and you get the usual 4x4 square made with the numbers 0-15. If you want multiplication instead, you can let (a,b,c,d)=(1,2,4,8) and (A,B,C,D)=(1,3,5,7), and you'll get magic product abcdABCD = 6720. I think this is the best choice -- you can use any values such that the sixteen products are pairwise distinct, and that seems the way to minimize it. Translating that back into numbers, you end up with 1 10 28 24 12 56 5 2 40 4 6 7 14 3 8 20 magic product: 6720 = 2^6.3.5.7 --Michael Kleber On 9/21/05, ed pegg <ed@mathpuzzle.com> wrote:
If you just use powers of 2, the minimal square with distinct entries has multiplicative constant 1073741824 (2^30). 55440 is much smaller. Is it the smallest?
--Ed Pegg Jr -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.