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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 7 to 11
Challenge Level

We received lots of good solutions to this task, so thank you to everybody who sent in their ideas!

Lots of children thought very carefully about what the best strategy was, but didn't notice that the petals you colour have to be next to each other. Benjamin from West Burton School in England explained his strategy clearly:

If you go first, colour in 2 petals so whether they colour in 1 or 2 petals, you can colour in the last one.

Thank you Benjamin, and thank you also to Anika from NAFL in India, the children at Greenwich House School in England, and Oak Class at Lythe School in the UK for sending in very similar explanations. Actually, there is a way for the second player to win...

The children at Waverley Primary School in the UK sent in lots of great ideas. Abbigail came up with a winning strategy for the second player:

Abbigail's method involved taking the second turn. If Player One took 1 petal, Player Two took the 2 opposite; if Player One took 2 petals, Player Two took the 1 opposite. This ensured that Player One would then be left with 2 petals available, but as they were not adjacent ('blocked' by Player Two), Player One could only shade 1, guaranteeing that the final petal would be left for Player Two. This proved to be an invincible strategy.

Well done, Abbigail! We had very similar explanations sent in by Florence and Mateo from Clifton College Prep School in England and by James from Twyford School in the UK. Thank you all for sending us your ideas.

Rye from Crudgington Primary School in the UK explained this in a slightly different way, and sent in a picture to show exactly what they meant:

We worked out that it's better to go as Player 2 as you can always win with the right move. If Player 1 starts with colouring in 1 petal, Player 2 will need to colour in 2 petals that are opposite (leaving 2 blank petals around Player 1's). If Player 1 starts with colouring in 2 petals, Player 2 will need to colour in 1 petal that is opposite (leaving 2 blank petals around Player 1 again). We quickly realised that we needed to split the last 2 petals left so that one player couldn't colour both in at the same time. Therefore it is always better to go second as you can create this scenario.

Thank you Rye!