# what is the formula for finding the number of squares on any sized chessboard, whether it is ow as 4 x 4 square or a 10,000 square chessboard(hypothetically)?i rally want a formula for any scenario...

what is the formula for finding the number of squares on any sized chessboard, whether it is ow as 4 x 4 square or a 10,000 square chessboard(hypothetically)?

i rally want a formula for any scenario with and sized square with smaller squares inside.

nathanshields | High School Teacher | (Level 1) Associate Educator

Posted on

1x1: 1

2x2: 1 + 2*2

3x3: 1 + 2*2 + 3*3

4x4: 1 + 2*2 + 3*3 + 4*4

nxn: `1^2+2^2+3^2+...+n^2`

I think I see the pattern, but we need a formula.  Imagine each square number as a physical square (made out of wood or something, like a collection of chessboards), and you stack them up, forming a pyramid.  Volume of a pyramid is `1/3Bh`, so we should see `1/3n^3` in our answer.

In fact, the sum of the first n square numbers is `(2n^3+3n^2+n)/6` , which is the formula you were after.