### Geoboards

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

If you had 36 cubes, what different cuboids could you make?

# Seven Flipped

## Seven Flipped

You have seven hexagonal-shaped mats in a line.
These mats all have to be turned over, but you can only turn over exactly three at a time.
You can choose the three from anywhere in the line.
A mat may be turned over on one move and turned back over again on another.

What is the smallest number of moves you can do this in?
Try with other numbers of mats.
Do you notice any patterns in your findings?
Can you explain why these patterns occur?

You might like to use the interactivity to have a go. Click on the three hexagons which you would like to turn over. The mats are red on one side (this side is face up to start with) and purple on the other.

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### Why do this problem?

This challenge is quite tricky, but it is a motivating context in which children can develop a logical, systematic approach.

### Possible approach

It would be useful to introduce the problem on an interactive whiteboard so that the whole group can be involved with deciding which mats to turn. Alternatively, cardboard mats coloured differently on each side could be used and pinned to a board. This initial whole group work will familiarise the children with the "rules" of the problem so that they will be confident to find the smallest number of moves in pairs.

It will be important for them to devise a recording system that they are happy with, and this is something that can be addressed in the introduction, for example by asking whether they would be able to repeat the moves they made. Encourage them to think about odd and even numbers of flips, and when they come to investigate other numbers of tiles, you might expect them to generalise for multiples of three at least.