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

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?

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?

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

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

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?

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?

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.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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

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

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?

How many models can you find which obey these rules?

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

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?

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

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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?

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

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.

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

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

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?

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

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

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

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

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?

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

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.

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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?

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?

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.

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

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

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?

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

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

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

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?

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

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?

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

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