It all lies in the method…
To count the total number of squares on a checkerboard, you have to consider squares of all sizes. The 1x1 and 8x8 squares are the easiest. There are 64 1x1 squares and a single 8x8 square.
For the 7x7 squares, they will leave one top or bottom row and one side column each. Thus, they each have to be stuck in one of the four corners. This means that there are four 7x7 squares.
For the 6x6 squares, the same principle can be used. Start in the bottom left corner and move one square right as you count. There are three that contain the last row, three that end at the penultimate row, and three that start at the top row for a total of nine.
For the 5x5 squares, you can think about it in the same way: there are 4+4+4+4=16.
At this point, you should be seeing a pattern. Let’s write out what we have.
1x1 squares: 64 = 82
2x2 squares: ?
3x3 squares: ?
4x4 squares: ?
5x5 squares: 16 = 42
6x6 squares: 9 = 32
7x7 squares: 4 = 22
8x8 squares: 1 = 12
The number of squares of each size is always a square number. We can conclude that there will be 52 4x4 squares, 62 3x3 squares, and 72 2x2 squares.
If we total them all up we get 1+4+9+16+25+36+49+64=204. A lot more than 64!
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