Dan Asimov wrote:
Then {additive magic squares of elements of N^k for some k} is in 1-1 correspondence with {multiplicative magic squares} -- just letting the components of v in N^k correspond to prime exponents.
I specifically thought this was *not* the case. In additive magic squares, for example, 1+1 = 2+0 -- well, that's a bad example since there's a duplicated number, but eg 5+1 = 2+4. But if you're trying to use the bits as bases for prime exponents, this is invalid: in binary you have 101 + 001 = 010 + 100, but if you count the bits on each side you get 102 versus 110. This is was I was fumblinly addressing with my comments about things being magic even if you forget how to carry. I'm not familiar with the Conway/Coxeter analysis Dan refers to, so perhaps there's a deep reason why normal magic squares -- well, some restricted subset of them -- automatically works on a bit-by-bit level. (Clearly not, say, for the 3x3 magic square on 0-8 -- there's only one entry with the 8s bit turned on!) I would guess that understanding when this happens, if it's not already known, is probably quite interesting. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.