In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?

The machine I use to produce Braille messages is faulty and one of the pins that makes a raised dot is not working. I typed a short message in Braille. Can you work out what it really says?

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

A game that demands a logical approach using systematic working to deduce a winning strategy

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?

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

This challenge extends the Plants investigation so now four or more children are involved.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Is it possible to use all 28 dominoes arranging them in squares of four? What patterns can you see in the solution(s)?

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

How many six digit numbers are there which DO NOT contain a 5?

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

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?

Can all but one square of an 8 by 8 Chessboard be covered by Trominoes?

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by. . . .

Which of these games would you play to give yourself the best possible chance of winning a prize?