We did not have many solutions sent in for this so if you come to this in 2014 let us know what you think.

Sydney worked hard at this problem. He wrote:

I tried various combinations of numbers of dots to make rectangles. I discovered by factoring each number of dots I could figure out how many rectangles I could make out of each number of dots.

That is very well expressed. In other words, by finding pairs of numbers that multiply together to make each number of dots, you can find out how many rectangles there are. Sydney continued:
For example:
So, all numbers make a skinny rectangle.
6 also makes a 2x3
1 dots make three rectangles: 1x12, 2x6, and 3x4.
8 a 2x4
10 a 5x2
12 a 2x6 and 3x4
14 a 2x7
16 a 2x8
18 a 2x9 and 3x6.

Pippa from Newbald Primary School sent in the following;

If you have 3 counters, you can make 2 rectangles. 1 x 3 and 3 x 1

If you have 6 counters, you can make 4 rectangles 1 x 6, 2 x 3, 3 x 2 and 6 x 1

If you have 18 counters, you can make 6 rectangles, 1 x 18, 2 x 9, 3 x 6, 6 x 3 , 9 x 2 and 18 x 1.

It's basically the times tables.
When you work out one answer  e.g 3 x 6 =18  just do the opposite to the numbers you are multiplying e.g 6 x 3 = 18.
And with other numbers of counters think of the times tables they are in.

Michael from Cloverdale Catholic in Canada wrote;

You can make about 6 rectangles:
1x18
2x9
3x6
6x3
9x2
18x1

Imagine these O'S are counters:
18x1 - OOOOOOOOOOOOOOOOOO

9x2  - OOOOOOOOO
OOOOOOOOO

6x3  - OOOOOO
OOOOOO
OOOOOO

3x6  - OOO
OOO
OOO
OOO
OOO
OOO

2x9  -
OO
OO
OO
OO
OO
OO
OO
OO
OO

1x18 -
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O

Thank you for these you certainly got good answers.