Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

What is the smallest number of coins needed to make up 12 dollars and 83 cents?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

What is the best way to shunt these carriages so that each train can continue its journey?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

How many trapeziums, of various sizes, are hidden in this picture?

What could the half time scores have been in these Olympic hockey matches?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!