Little Problem about rectangles

[doggie_Walkes]

Hey this is just a fun question that my teacher said. But haven't got a clue.

Suppose you draw a n x n grid on a piece of paper. how many squares could you draw in the diagram? how many rectangles can you contain.

Its something to do with binomial coefficients, what you guys think?

 

[futurebird]

"how many rectangles can you contain."

What do you mean by this?

 

[jegues]

I'm not sure if this is correct but I'll throw it out there anyways.

If we let n = 1, how many squares are we going to get? 1 right?

If we let n = 2, how many squares are we going to get? 4 right?

If we let n = 3, how many squares are we going to get? 9 right?

Hmmmm... Can you see the pattern developping?

As far as the rectangles question, I'm not entirely sure what you mean by that, we could make infinitely small rectangles couldn't we? There needs to be more clarification on that part of the question.

EDIT: If a rectangle can only be made from combining "WHOLE" squares on our grid paper then the smallest rectangle would be generated from combining 2 squares, so if you're asking how many rectangles we can contain it would simply be HALF the number of squares.

 

[Dick]

I'm sure the question just means you make the squares or rectangles by combining whole 1x1 squares. For n=2 you get 5 squares, right? 4 1x1 and 1 2x2. n=3 I get 14.

 

[futurebird]

No, for nXn there are

\sum_{i=0}^{n} (n-i)^2.

Think of a 2x2 grid. there are 4 little squares and one big one, on a 3x3 grid you have 9 little squares, 4, 2x2 squares and 1 big one... see the pattern?

(Though, I still don't know what the thing about rectangles is getting at.)

 

[Dick]

Let's try not to give the whole thing away here, but a rectangle is defined by choosing any two grid points which aren't in the same row or column. It's a combinatorics problem.