Couldn't sleep so started thinking about solving it.
Think of each square as a 'cell'. This is just to prevent confusing 'squares' on the board with 'squares form by squares'. The rows are labelled 1 to 8 with A being the closest row to you. The columns are labelled A to H with A being the closes row to you.
In an 8x8 arrangement of cells there are 8x8 squares consisting of a single cell ie 64
There are also a number of squares formed by groups of 2x2 cells. Starting at the bottom left (Cell A1) of the board imagine a 2x2 square. Move to the next cell to the right (B2). Use this cell as the 'bottom left' corner of another 2x2 square.
Repeat until you run out of additional cells to the right to complete the 2x2 square. There will have been 7 squares formed, each 2x2.
The same logic applies to moving up the board ie 7 squares along each row until running out of additional cells after Row 7.
This means the total number of 2x2 squares must be equal to 7x7 = 49.
Using the same logic you will find that for each step UP in the number of cells forming a square ie 2x2, 3x3, 4x4 etc there is a corresponding reduction in the number of squares that can be formed. 3x3 cells on an 8x8 board will only allow 6x6 squares to be formed, 4x4 cells gives 5x5 squares etc etc
So, the total answer is
8x8 + 7x7 + 6x6 + 5x5 + 4x4 + 3x3 + 2x2 + 1x1 =
64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 =
204!
. .Christ, the things one thinks about with insomnia!
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