What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

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.

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

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

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.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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 put the numbers 1-5 in the V shape so that both 'arms' have the same total?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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

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?

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.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

An investigation that gives you the opportunity to make and justify predictions.

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!

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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.

Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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

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

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

You have 5 darts and your target score is 44. How many different ways could you score 44?

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?

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?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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?

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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?

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?

Investigate the different ways you could split up these rooms so that you have double the number.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Can you make square numbers by adding two prime numbers together?