# rectangles

• May 11th 2010, 10:29 AM
ihavvaquestion
rectangles
how many rectangles (including squares) be found in the following diagram? you must explain why your calculations are actually counting the possible rectangles or squares in this diagram. Its a 4x8 grid...i did not know how to draw it w/o attaching it. i think it is an nCr problem, I'm just not quite sure
Code:

 :------------------------------------------: :    :        :        :      :    :    :    :    : :------------------------------------------: :    :        :        :      :    :    :    :    : :------------------------------------------: :    :        :        :      :    :    :    :    : :------------------------------------------: :    :        :        :      :    :    :    :    : :------------------------------------------:
• May 11th 2010, 10:51 AM
undefined
Quote:

Originally Posted by ihavvaquestion
how many rectangles (including squares) be found in the following diagram? you must explain why your calculations are actually counting the possible rectangles or squares in this diagram. Its a 4x8 grid...i did not know how to draw it w/o attaching it. i think it is an nCr problem, I'm just not quite sure

:------------------------------------------:
: : : : : : : : :
:------------------------------------------:
: : : : : : : : :
:------------------------------------------:
: : : : : : : : :
:------------------------------------------:
: : : : : : : : :
:------------------------------------------:

You should enclose the diagram in [ code][ /code] tags (I put in extra spaces so that they will be visible). The spacing is all messed up.

Here's an example of something in code tags:

Code:

+----+ |    | +----+
• May 11th 2010, 01:03 PM
ihavvaquestion
ok thanks. so is this an nCr problem? i was thinking that there are 32C3, but i know that is not right
• May 11th 2010, 02:18 PM
undefined
Quote:

Originally Posted by ihavvaquestion
ok thanks. so is this an nCr problem? i was thinking that there are 32C3, but i know that is not right

Okay so the diagram still doesn't look right on my system, but I think the question is equivalent to asking for how many rectangles are in the following figure,

http://i41.tinypic.com/vhx2is.png

which is called a $9 \times 5$ grid graph, or $G_{9,5}$.

This question is common in recreational mathematics, e.g., puzzle books and puzzle websites. You have to find a systematic way to count them all. My recommendation: Fix the upper left corner first, then fix the height, then see how many rectangles you can find. Iterate through all possible upper left corners and all possible heights given those upper left corners. Look at smaller examples if it's easier, or to convince yourself that you have a valid method.
• May 11th 2010, 04:06 PM
awkward
Consider the horizontal and vertical lines which determine the sides of a rectangle.

There are $\binom{9}{2}$ ways to pick the vertical lines and $\binom{5}{2}$ ways to pick the horizontal lines.

So...
• May 11th 2010, 04:18 PM
undefined
Quote:

Originally Posted by awkward
Consider the horizontal and vertical lines which determine the sides of a rectangle.

There are $\binom{9}{2}$ ways to pick the vertical lines and $\binom{5}{2}$ ways to pick the horizontal lines.

So...

I knew the answer in terms of triangle numbers and never thought to look at it that way. Thanks!
• May 11th 2010, 05:17 PM
ihavvaquestion
ah that makes sense looking at it that way...360 rectangles...thanks