If you like playing with chess boards, try The EIGHT QUEENS Puzzle! 
1
First of all the whole board makes one BIG square. 
+ 4
Then there are four slightly smaller squares (measuring 7x7). 
We've marked where the corners go on the main board using numbers 14.
We've outlined where the first square goes in red. 
+ 9
Then there are nine squares measuring 6x6. 
We've marked where the corners go on the main board using numbers 19.
We've outlined where square number 2 goes in blue. 
When you add a list of square numbers you get the
The formula for the number of balls in a square pyramid = [ 2n^{3} + 3n^{2} + n ] ÷ 6where n = the number of layers of the pyramid.

So how many squares of different sizes have we counted so far? It's 1 + 4 + 9 + ... .
You might have noticed these are the square numbers. 1^{2}=1 2^{2}=4 3^{2}=9
Here's the next smaller size of square. How many can we fit on a chessboard? It's the next square number!

Here are the other possible squares. 

Numbers of each square that will fit on the board. 