Problem Solving

I'm working on a word problem that I can't seem to wrap my head around. Can someone please breakdown how to solve this? Thanks!

How many square regions are there on a chess board ( a square region can have sides of lenght one through eight) ?

What about on an n X n chess board?

There is a formula to calculate the no of squares here...
if there are n rows and n columns on a board then no. of squares = n^2 + (n-1)^2 + (n-2)^2 + ...+ 2^2 + 1^2

so for a 4 x 4 board there are 16 + 9 + 4 + 1 = 30 squares
similarly for 3 x 3 board there are 9 + 4 + 1 = 14 squares

hope this helps...

You don't need to memorize a formula. Instead, reason through it:

Let's use an 8x8 board.
How many 8x8 squares are there? 1

How many 7x7 squares are there?
Well, start with one at the top left. We could make new ones by moving it:
down 1 unit
over 1 unit
down and over 1 unit
Total: 4
In other words, the top-left corner of the 7x7 can be in any of the 4 squares defined by a little 2x2 square.

Similarly, for 6x6 squares, the top left corner can be in any of the 9 squares defined by a little 3x3 square.

You can see the pattern emerging: 1 + 4 + 9 ...