Last night I got around to:
Aa Bc Cd Db Cb Dd Ac Ba Dc Ca Bb Ad Bd Ab Da Cc
...and then I suggested
(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[...]
But it's not; I was seduced by the siren song of powers of two. Much better to use (a,b,c,d) = (1,2,3,6) and (A,B,C,D) = (1,4,5,7), so abcdABCD = 7! : 1 12 30 14 10 42 3 4 21 5 8 6 24 2 7 15 magic product = 5040 = 2^4.3^2.5.7 As I was falling asleep I thought I had an information-theoretic proof that this was optimal. [It started by pointing out that (1) in this version, you can specify each entry using only 5 bits -- whether or not to multiply each of 23457 -- and saying that 5040 was the smallest product you could pack 5 bits into; and (2) Rich already optimized 4 bits and got 14400, while 6 bits or more is again going to be worse.] But things that were clear last night are fuzzy this morning, and the above doesn't seem to prove anything at all. Maybe someone else will un-muddle things. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.