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?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

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?

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?

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?

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.

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

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?

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

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.

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

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.

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

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.

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

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?

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?

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?

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?

How many models can you find which obey these rules?

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

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

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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

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 stack these rods, following the rules?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

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

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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

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?

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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

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?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

The pages of my calendar have got mixed up. Can you sort them out?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

In this matching game, you have to decide how long different events take.

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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.