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?

I think this got off the queue without its difficulty changed. I think it should be 2/5.

The way I prove how to get the number of squares in a board is like this:

For a 24x24 square on a 24x24 board, there is 1 bottom right square such that this will work.

For a 23x23 square on a 24x24 board, there are 4 bottom right squares on the 24x24 board that could be the bottom right square of the 23x23, without the top and left edges of the 23x23 square running off the board.

This would continue on, so the number of squares on a n-by-n board is the sum of n² down to 1².

The sum of 24² down to 1² is 4900, and the square root of this is 70, so 70 is the answer.

Posted by Gamer on 2003-06-30 02:47:45