# Thread: Counting sides

1. ## Counting sides

A rectangle that measures 4cm by 6cm is divided into 24 squares with sides 1cm in length. What is the total number of 1cm long sides in those 24 squares? (If 2 squares share a side, the side should be counted only once).

Now I know the answer is 58 and I drew a sketch and counted the sides but I was wondering if there was any shortcut to this problem that I was missing..? Setting up any kind of expression or equation or maybe a combinations problem of some sort?

2. Let use a vertical layout like this (for explanation):

****
****
****
****
****
****

First count all the right and bottom sides of each square: 24 * 2 = 48. There is no overlap in this counting.
However, we missed some sides with this count. Which ones? The entire top row and left column.
Well, the number of edges we did not count is:
# on top + # on left = 4 + 6 = 10

Grand total is: 48 + 10 = 58

So from this the general solution for number of sides in a n by m block is:
2nm + n + m = n(1 + m) + m(1 + n)

This suggest thats an alternative way is to count:
#horizontal edges + #vertical edges = n(1 + m) + m(1 + n)

3. Hi snow--your solution definitely seems to work but I think there is a column of 6 missing Thanks!

4. Hello, sarahh!

A rectangle that measures 4cm by 6cm is divided into 24 squares with sides 1cm in length.
What is the total number of 1cm long sides in those 24 squares?
(If 2 squares share a side, the side should be counted only once).

You made a sketch but you didn't see any pattern, did you?

Code:

* - * - * - * - * - * - *
|   |   |   |   |   |   |
* - * - * - * - * - * - *
|   |   |   |   |   |   |
* - * - * - * - * - * - *
|   |   |   |   |   |   |
* - * - * - * - * - * - *
|   |   |   |   |   |   |
* - * - * - * - * - * - *

The unit squares are in 4 rows and 6 columns.

Imagine these 24 squares formed with matches.
How many matches are used?

We see that there are 6 matches in each row.
And there are 4 + 1 rows.

Hence, there are: .6(4+1) horizontal matches.

We see that there are 4 matches in each column.
And there are 6 + 1 columns.

Hence, there are: .4(6 + 1) vertical matches.

We can generalize this problem.
Suppose there are R rows and C columns.

Then there are: .C(R+1) horizontal matches
. . . . . . . .and: .R(C+1) vertical matches.

Total: .$\displaystyle C(R+1) + R(C+1) \;=\; R + C + 2RC$ matches.

5. Originally Posted by sarahh
Hi snow--your solution definitely seems to work but I think there is a column of 6 missing Thanks!
There is a row of 6 and a column of 4 missing. Which is why a 10 (= 6 + 4) is added at the end.

6. Just for fun, here is another approach-- first do it the wrong way, then patch it up.

There are 24 squares in all, each with 4 sides, and each side is shared between two squares, so there are

$\displaystyle \frac{4 \times 24}{2} = 48$ sides.

But oops, the sides along the perimeter of the rectangle (2 * (4 + 6) = 20) are not shared! So we need to add those back in.

$\displaystyle 48 + \frac{20}{2} = 58$