### 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?

# Wag Worms

##### Stage: 2 Challenge Level:

Many people produced good results and it was interesting to see the different ways the problem was interpreted.

Did you decide that you can add another cube onto one that you have already added to or not? So, in the following, can the latest blue cube go in the places pictured in the 2nd and 3rd diagrams?

Michael wrote in to say:

There were five places where a cube could be added on to each of the five Wag Worms, and $5\times5 = 25$. So, there are $25$ different types of Wag Worm in its third year.

The Gateway School sent in two solutions of $25$ as well.

Rowena wrote the following good explanation

I know that a cube has $6$ faces. I looked at the Wag Worm in its second year, you can add a cube to all of the $6$ faces except the one with eyes, so there are $5$ possible Wag Worms.
For the Wag Worms in their third year, I looked at adding a cube to the 'non-eye cube' and the 'eye cube'.
For the 'non-eye cube' there are $5$ visible faces (one is attached to the 'eye cube') so I could make $5$ different Wag Worms.
For the 'eye cube' again there are $5$ visible faces, but one has eyes, so I could have $4$ places to put the cube.
This is $5+4=9$ possible Wag Worms for each of the 5 Worms in their second year. So in total there would be $5\times9 = 45$ different Wag Worms in their third year.

Emma and Oscar went further and tried the fifth year of growth. Here are some of their results: