Find the number of squares in a square grid

[bloodhound]

Ok. you have a n by n grid of squares. ie. made out of n^2 small squares.

 

all u have to do is find the total number of squares of all sizes.

 

If u want to , you can extend the problem into a rectangle of n by m squares .

 

Also find the number of possible rectangles(rectangles also includes squares)

[jordan]

Ok. I think I have the answer. It is difficult to explain though without proper math symbols on the keyboard. At least for the squares part, the answer is

 

n^2+(n-1)^2+(n-2)^2...

 

That just covers squares though.

[wolfson]

For rectangles, here's a way of thinking about it which is helpful.

Drawing a rectangle on the board is equivalent to drawing two vertical lines and two horizontal lines, and then colouring in the rectangle enclosed by them.

Think about how many choices you have about where to draw each line.

 

Jordan try and integrate the problem to find an nth term.

[jordan]

Again with the integration, Wolfson. I have the answer to this one, just don't know how to write it properly. Mabey integration can help. If you feel like explaining it I'll be here or on AOM's. Oh, how does my squares equation look?

[wolfson]

Good, i will add a bit tomorrow, oh and ive answered your math question on AOM.

[bloodhound]

to jordan: yeah thats rite

 

the answer u gave is equivalent to sum of k^2 from 1 to n. which gives u n(n+1)/2

 

to wolfson: ur rite on spot as well.

 

basically a rectangle consists of two vertical and horizontal lines. there are (n+1) horizontal and vertical lines in a square grid

 

there are ⁿ⁺¹C₂ ways to choose 2 lines out of n+1 . also for each of those u have the same amount of ways u can choose two vertical lines.

 

there fore u have the total no. of retangles = ⁿ⁺¹C₂ squared.

which i think is equal to (n+1)!/2!(n-1)! squared. = n^2(n+1)^2/4

 

now try the same thing to find the nu. of squares and rectangesl in a n by m grid