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.