How many squares are there on a chess board?
Oh dear  this simple question leads to some big arguments between the chess pieces!
The board has 8 squares along the bottom and 8 squares up the side, so most people would say the answer is 8x8 =64.
BUT suppose you count the BIG square as one, and then you also want to count up all the smaller squares  how many squares are there in total?


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
Square Pyramid Numbers.
The formula for the number of balls in a square pyramid =
[ 2n^{3} + 3n^{2} + n ] ÷ 6
where n = the number of layers of the pyramid.
The number of squares on a chessboard = the 8th pyramid number.
So if you make n=8 in the formula you'll get the answer!
Find more fantastic formulas in THE PERFECT SAUSAGE. You can even find out how many rectangles are on a chess board using triangle numbers!
(The answer is T_{8} x T_{8} = 36 x 36 = 1296).

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!
4^{2} = + 16


So the total number of squares on the chess board = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204
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