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


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

Multilink Cubes

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


Age 5 to 11 Challenge Level:


You've maybe come across a challenge of putting five squares together in different ways - often called pentominoes.

So here are five squares and four ways of putting them together - there are obviously more ways. (You could try the Penta Place problem if you haven't seen this before.)
You may even have done a similar challenge using triangles (if not have a go at our Tri-five problem). Like this:
But the challenge for today is to consider the same idea using hexagons.
So, the challenge is:

What different shapes can you make using five hexagons?

Be careful that you do not have any the same as each other. For example, these two look different but are in fact identical:


Please send in all the different shapes you can get and let us know how you made sure you had found them all.

Perhaps you also have some ideas about how you may go further with the question "I wonder what would happen if I ...?"

Why do this problem?

This activity is accessible to a wide variety of pupils. At one level, learners can explore ways of putting the five shapes together by 'playing' with hexagon pieces. At a higher level, more experienced or older pupils can be expected to go about the activity in an organised, systematic way. In addition, this problem encourages learners to use visualisation, and to apply their understanding of reflective and rotational symmetry.

Possible approach

You may like to start by giving pupils the experience of this problem using five squares (pentominoes). Rather than asking them to make the different shapes, you could give them the set of twelve pentominoes and arrange them in different ways so that they are convinced there aren't any missing. This will help them to develop a system for working on the hexagon version.

You could introduce the challenge by creating some hexagons to be dragged around the interactive whiteboard screen, or you could cut large shapes out of card and fix them to the board or wall. Encourage a few children to make their own arrangements for all to see so that everyone is clear about the task before giving time for pairs to investigate further. Children will find it useful to cut out hexagons from this sheet (larger shapes) or perhaps to colour in arrangements on this sheet (smaller shapes).

You may need to have a mini-plenary at an appropriate moment to discuss shapes that may be reflections or rotations of each other. At this point, having the interactive whiteboard would be helpful so that you can demonstrate rotations or flips easily (although this can be done by cutting out shapes or joining hexagons together so they can be manipulated physically).

In the plenary, you could pool all the different shapes that the group has found. You could then either invite some pairs up to explain their own ways of working systematically, or you could ask the class to order the shapes that are displayed so that any missing ones can be identified.

Key questions

Are any of your shapes the same?
Have you done this in some way so that you know what you are going to try next?
How are you making sure you do not have the same one twice?
How are you recording these?

Possible extension

One possibility is for pupils to explore what happens with more hexagons. Alternatively, you could encourage them to ask "I wonder what would happen if I ...?" and to go down their own line of enquiry.

Possible support

An appropriate drawing program on the computer may be a useful tool for some pupils. Others will need to have hexagons to arrange.