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A group of children from Manorfield Primary School, Stoney Stanton sent in lots of ideas:

S.B. and N.L. produced the following table of results:

Number Shape Number of pebbles on side Area of shape Perimeter of shape
1 Square 2x2 1cm$^2$ 4cm
2 Rectangle 2x3 2cm$^2$ 6cm
3 Square 3x3 4cm$^2$ 8cm
4 Rectangle 3x5 8cm$^2$ 12cm
5 Square 5x5 16cm$^2$ 16cm
6 Rectangle 5x9 32cm$^2$ 24cm
7 Square 9x9 64cm$^2$ 32cm
8 Rectangle 9x17 128cm$^2$ 48cm

PATTERNS AND FORMULAE
It was said in whole class discussion that the pattern for the size of the shapes was:
Squares: each side is the same length as the longest side of previous rectangle
Rectangle: one side is the same length as the side of the previous square, the other side is a "new length"

A.H and E.R also said:

The formula for the area = $2$ to the power of $(n-1)$.
The pattern for the new side of the rectangles is $+1$, $+2$, $+4$, $+8$, $+16$ ... (it doubles)
The pattern for the perimeter is $+2+2$, $+4+4$, $+8+8$, $+16+16$ ... (it doubles)

Now, Alice from St Thomas' Church of England Primary School, wrote;

On B (how many pebbles there are round the edges), the pattern starts on 4 adds 2 which makes  6 adds another 2 which is eight then adds 4 which makes 12 adds 4 again (16) adds 8  (24) adds 8 adds 16, and so on.
So it adds one thing twice then doubles the adding number and starts to do that again.

Amelia from NLCS sent in her ideas;

First double forward then double sideways and so on
Pebbles added: 2,3,6,10,20,36,72
Pebbles Outside: 4,6,8,12,16,24,32
Squares: 1,2,4,8,16,32,64,128

Well done all of you - you obviously worked hard on this problem.