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what is the formula for finding the number of squares on any sized chessboard, whether...

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boogey--lee | Student, Grade 9 | eNoter

Posted January 12, 2012 at 6:48 AM via web

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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.

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

Posted January 12, 2012 at 8:06 AM (Answer #1)

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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.

 

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