At 09:07 PM 5/30/2003, you wrote:
I've never seen this trick before.  What properties of the subsquares make this
trick work?  Would it be possible to perform this trick with a square that has
distinct entries?  Entries 1 through 64?

http://www.mdani.demon.co.uk/stunt/may98s1.htm

I think it is based on Latin squares.  When you pick a 4x4, and do the procedure, you select one and only one number from each row and column.  Any such set of four numbers for a given 4x4 sum to the same number.