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.