the square squared puzzle got on a quest.
here's some interesting factoids.
the problem of squaring the square with smaller squares, and finding the minimum number of squares necessary to do so is quite the challenge. it is currently unknown if the minimum squares necessary to tile an m x n square is always also the minimum for a k*m x k*n square.
squaring the square in a "no where neat" fashion; that is, such that two squares of the same size don't completely align, can be quite fun. i have yet to find an example where every square, both the outer square to be filled and all inner squares, are all prime size.
the smallest packing size for all integers can also be quite the challenge. it's known for 1-51.
just the tip of the iceberg I'm sure.