how many squares

[Void]

'lo everyone.

I was struck by a picture on facebook some time ago. it was the following:

with the question 'How many squares do you see'

Not too hard of a challenge. What did occur to me though was whether or not there would be a formula for these squares. So I started the math, and found the formula. For those interested, it is:

Spoiler

C(n)=(1/3)n^3 + (1/2)n^2 + (1/6)n

where C is the number of squares, and where n is the number of squares on a side, e.g. in the picture it's 3.

I also found that:

C(n) = n^2 + C(n-1)

but that of course wouldn't help if you just wanted to know C(n) for n=5

Turns out, it was already on the internet somewhere, so my actions were in vein. Or were they? I kinda enjoyed this adventure, and I would like to find more of these. But I can't. Can any of you maybe direct me to similar problems?

EDIT: I don't know if this is the correct subforum :S

 

[Vrecknidj]

X = total # of squares
n = number of squares per side

X = (n)(n) + (n-1)(n-1) + (n-2)(n-2) + ... + (1)(1)

Your example

14 = (3)(3) + (2)(2) + (1)(1)

 

[Vrecknidj]

∑(x)^2 = (n)(n + 1)(2n +1)/6