### 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?

# Counting Cards

##### Stage: 2 Challenge Level:

Maggie of St Anne's School sent the following partial solution, well done Maggie.

You have to arrange the cards in a certain order . For example, the order for 1 2 3 4 5 would be 2 4 1 3 5
O...N...E
1...2...3

T...W...O
4...5...1

T...H...R...E...E
2...4...5...2...4

F...O...U...R
5...2...5...2...

F...I...V...E
5...5...5...5

So the cards would be arranged with

Ace in position 3
Two in position 1
Three in position 4
Four in position 2
Five in position 5

There were a number of other anonymous solutions and a solution from Andrei of School 205 Bucharest. Here is one method suggested by one person:

Let's say you have 10 cards - imagine 10 positions that represent the order of the cards in the pack

#### 10

Then you can place each card in turn in each position - using up the space so:

#### 10

 O N 1 T W 2 T H R E

but you run out of room before you can spell out THREE so you go back to the beginning and 3 will go in position 1. The you start spelling FOUR but you have to jump over position 3 because it has a 1 in it and position 6 because it has a 2 in it, so 4 ends up in position 7:

#### 10

 3 F 1 O U 2 4

So you end up with:

#### 10

 3 5 1 8 10 2 4 6 7 9

This method works for any number of cards and whether you use their names or their values.