The number of 1x1 squares on a chessboard. The number of squares on a chessboard is 204

1, 8x8 square 4, 7x7 squares 9, 6x6 squares 16, 5x5 squares 25, 4x4 squares 36, 3x3 squares 49, 2x2 squares 64, 1x1 squares

> The number of squares on a chessboard is 204

The number of squares on a chessboard is infinite, if you don't limit yourself to edges already drawn for you.

Yes, and of course you should really generalize that so that the number of squares on a board with nxn single squares is sum(i=1..n) i*i

This thread is pedantic.

By thy way:

  sum(i=1..n) i*i = n/6 + n^2/2 + n^3/3

Proof?

This proof helps: http://pirate.shu.edu/~wachsmut/ira/infinity/answers/sm_sq_c...

From that: sum bla= n(n+1)(2n+1)/6, then you go distribute

Nice. How did I get the result in the first place?

I know that this sum can be expressed as a cubic polynomial. So I just evaluated the sum at four data-points (n=0,1,2,3) and used those to determine the four coefficents by solving a linear equation system.

Turns out that drawing a picture really helps: http://news.ycombinator.com/item?id=101532

You can usually come up with a clever picture only after you understood the problem - not before.

That's very keen observation indeed!

that's very interesting.