Thread: CHESSBOARD PIECE OF WORK

1. CHESSBOARD PIECE OF WORK

Hi,

I am helping my son do some maths homework. We have worked out that there are 204 squares on a chessboard, and we have worked out that this is relative to square numbers.

How can I express a formula tidily? Is there a function that is similar to Factorial but using addition? EG: x=n²+(n+1)²... (n+7)² simplified to a tidier function (where n = 8)?

The formulae for a chessboard is x=1²+2²+3²+4²+5²+6²+7²+8², so how can I use this using n? eg: something like x=(n²)! but without factorial and with the appropriate function?

2. I am not really sure I follow your question.
But it is well known that $\sum\limits_{k = 1}^N {k^2 } = \frac{{N\left( {N + 1} \right)\left( {2N + 1} \right)}}{6}$.

If $1 < J < N$ then $J^2 + \left( {J + 1} \right)^2 + \cdots + \left[ {J + (N - J)} \right]^2 = \sum\limits_{k = J}^N {k^2 } = \sum\limits_{k = 1}^N {k^2 } - \sum\limits_{k = 1}^{J - 1} {k^2 }$.

3. Thanks, I'll try to re-phrase...

x=(n²)! (where n = 8)
expands to...
x=1²*2²*3²*4²*5²*6²*7²*8²

Can I use any function to do similar but using addition instead of multiplication eg...
x=1²+2²+3²+4²+5²+6²+7²+8²
so x = 204

Basically, a formula that works out how many squares a grid of any given size contains. EG: 8*8 grid has 204 possible squares within it (1², 2², 3², 4², 5², 6², 7², 8²)

Is this any clearer?

4. Originally Posted by danielmegson
x=1²+2²+3²+4²+5²+6²+7²+8²
so x = 204

Basically, a formula that works out how many squares a grid of any given size contains. EG: 8*8 grid has 204 possible squares within it (1², 2², 3², 4², 5², 6², 7², 8²)

Is this any clearer?
It was clear the first time for the sum.
It is after all a sum and not a product.
I gave you the formula for the sum.
This is a well known problem and you have been given the solution twice on this board.

5. Thanks, sorry I'm a bit rusty at mathematics - been a long while! Thanks for your help!