In connection with Story #85, an MCMC take on Magic Squares, I've uploaded com5d to the site. Use it, play with it, constuct magic squares of sizes 4 x 4, 5 x 5, 6 x 6, perhaps higher. This is achieved via the MCMC for f(x) = \exp[-\lambda Q(x)] / constant, with positive \lambda. You may also play with *negative \lambda*, to encourage outcomes as far as possible from magic-ness: sums of rows, columns, diagonals, far away from the magic number (34 for 4 x 4, 111 for 6 x 6, etc.). The exercise is then to find the Bayesian MAP, maximising Q(x) as score: the worst ever squares.
I think it's pretty hard to do this by pure math -- so run chains and see what happens. My current guesses, for 4 x 4 and 6 x 6 (after tne minutes of playnig) are 114 and 540. If you can beat these numbers, tell me, and I'll give you tyve kroner (les Rudolf Nilsen).