How many squares are there on an 8x8 grid?

Answer 1

1²+2²+3²+4²+5²+6²+7²+8²=204 Size Of square Number of squares --------------- ----------------- 1 x 1 8^2 = 64 2 x 2 7^2 = 49 3 x 3 6^2 = 36 4 x 4 5^2 = 25 5 x 5 4^2 = 16 6 x 6 3^2 = 9 7 x 7 2^2 = 4 8 x 8 1^2 = 1

I can make it slimier by using Faulhaber's formula

First Proof

Second Proof

(On the LHS, all the terms will get cancelled (Except (n+1) and 1))

Third Proof

The equivalence between the sum of squares and the cubic polynomial may also be shown by a double counting proof in which one counts in two different ways the number of ways to choose three numbers x, y, and z from the set {1, 2, 3, ... n + 1}, in such a way that z > x and z > y.

First, we fix z and consider the number of ways of choosing x and y. If z = 1 then there are no values of x and y that satisfy the inequality, if z = 2 then the only possible choice is x = y = 1, if z = 3 then x and y may independently be chosen to be either 1 or 2, and in general once z is chosen there are (z − 1)2 ways of choosing x and y. Hence the total number of ways of choosing x, y, and z is.