### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

### Tri-colour

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

### Permute It

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

# Euromaths

##### Stage: 3 Challenge Level:

Sarah Dunn, Madras College, St Andrew's, Scotland and Soh Yong Sheng, Raffles Institution, Singapore both solved this in the same way. You write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in this array.

 E 0
 U 1
 R 1
 O 1
 M 1
 U 1
 R 2
 O 3
 M 4
 A 5
 R 1
 O 3
 M 6
 A 10
 T 15
 O 1
 M 4
 A 10
 T 20
 H 35
 M 1
 A 5
 T 15
 H 35
 S 70

We draw a grid denoting the number of moves possible to reach each place. The number of possible routes are calculated by adding the number of the gridplace on the left and top, and if it is on the extreme left or top then there is only 1 route to get there. The number added will be correct as the square can only be accessed through these squares. There are altogether 70 possible ways.

Can you generalise this result to a 6 by 6 square, or a 7 by 7 square ... or an n by n square? Have you seen this pattern before? You may like to try a 6 by 6 array written in a slightly different formation.

 M A A T T T H H H H E E E E E M M M M M M A A A A A T T T T I I I C C S