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 shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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?

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

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

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

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?

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

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?

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

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

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

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

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?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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

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

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.

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

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.

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?

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

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

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?

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?

How many models can you find which obey these rules?

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

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?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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?

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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 work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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

My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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.

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?

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. . . .

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?