Imagine a 24-by-24 chessboard. Now suppose you started counting all of the "sub-squares" on that board, squares of lengths 1 through 24 found by tracing the sides of the squares of the big board. To remind you how many sub-squares you've counted, you make a pile of little squares of all equal size (which you just happen to have lying around), one little square for each sub-square.

It turns out that these little squares can be put together, edge to edge, to form an even bigger chessboard.

What is the length of each side of the giant chessboard?

70

To see how many subsquares of a certain size there are, start with 23. That you can start in the upper left corner and then shift once right, once down, then left for a total of 4. For 22, you can do the same going twice right and twice down, including the first spots, it's a 3x3 move or 9. So the solution is simply the sum of 1 through 24 squared which comes to 4900, the squareroot of which is 70. Not too tough.

Posted by Lawrence on 2003-08-27 02:01:54