How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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?

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?

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?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

How many models can you find which obey these rules?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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.

These practical challenges are all about making a 'tray' and covering it with paper.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

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?

Can you find all the different ways of lining up these Cuisenaire rods?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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?

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?

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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.

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

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