Combinatorics

problem
Snowman

Age

14 to 16

Challenge level

All the words in the Snowman language consist of exactly seven letters formed from the letters {s, no, wm, an). How many words are there in the Snowman language?
problem
Cube paths

Age

11 to 14

Challenge level

Given a 2 by 2 by 2 skeletal cube with one route `down' the cube. How many routes are there from A to B?
problem
Penta colour

Age

14 to 16

Challenge level

In how many different ways can I colour the five edges of a pentagon so that no two adjacent edges are the same colour?
problem
Scratch cards

Age

14 to 16

Challenge level

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?
problem
Tri-colour

Age

11 to 14

Challenge level

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?
problem
Painting cubes

Age

11 to 14

Challenge level

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?
problem
W mates

Age

16 to 18

Challenge level

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.
problem
Knight defeated

Age

14 to 16

Challenge level

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board for any value of n. How many ways can a knight do this on a 3 by 4 board?
problem
Magic W

Age

14 to 16

Challenge level

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
problem
Olympic magic

Age

14 to 16

Challenge level

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?