Last year at this time, Chess master Garry Kasparov was having trouble with IBM's Deep Blue program. Here's an easier chess problem for the rest of us. There are 64 single squares on a normal 8 by 8 chessboard. But the chessboard itself is also one big 8 by 8 square, there are lots of overlapping 2 by 2 squares on the board, and there are many others. So how many squares are there altogether, of any integer size? Hint: start at the upper left corner and count different size squares across and down, looking for a pattern. What about an arbitrary size n-by-n or n-by-m chessboard?
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