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?

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.

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?

What happens when you try and fit the triomino pieces into these two grids?

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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?

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?

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

How many models can you find which obey these rules?

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

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

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?

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 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.

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

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?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

My coat has three buttons. How many ways can you find to do up all the buttons?

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?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

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

Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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?

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

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?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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.

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

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.

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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?

What could the half time scores have been in these Olympic hockey matches?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

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

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

How many different rhythms can you make by putting two drums on the wheel?

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?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

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

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

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?

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....?