- How many squares are there on a chessboard or checkerboard?? (the answer is not 64)
Can you extend your technique to calculate the number of rectangles on a chessboard.
This another puzzle that was also e-mailed to me through this website. My instinct was that the answer was just a lot, but i thought about it and the solution is actually fairly simple...
The first thing is why the answer is not just 64.<All the red squares in the above picture would count as valid squares, so we are asking how many squares of any dimension from 1x1 to 8x8 there are on a chess board.
The key is to think how many positions there are that each size of square can be located...
A 2x2 square, for example, can be located in 7 loactions horizontally and 7 locations vertically. ie in 49 different positions. Consider the table below...
| size | horizontal positions | vertical positions | positons |
| 1x1 | 8 | 8 | 64 |
| 2x2 | 7 | 7 | 49 |
| 3x3 | 6 | 6 | 36 |
| 4x4 | 5 | 5 | 25 |
| 5x5 | 4 | 4 | 16 |
| 6x6 | 3 | 3 | 9 |
| 7x7 | 2 | 2 | 4 |
| 8x8 | 1 | 1 | 1 |
| total | 204 |
In total there are 204 positions. this is the sum of the number of possible positions for all the
different sized squares.
