Squares on a Chessboard

How many squares are there on a chessboard? (Hint: The answer is not 64)

Answer

204 squares.

The chessboard is made up of 8×8 squares. We have to determine how many squares of any dimension from 1×1 to 8×8 there are on a chess board.

The key is to determine how many possible positions there are for each type of square. A 2×2 square, for example, can be located in 7 locations horizontally and 7 locations vertically. eg. in 49 different positions.

sizehorizontal positionsvertical positionssquares 
1x18864
2x27749
3x36636
4x45525
5x54416
6x6339
7x7224
8x8111
Total204

In total there are 204 positions. this is the sum of the number of possible positions for all the different sized squares.