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?

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?

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?

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

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?

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?

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

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?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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

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

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

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 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?

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.

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?

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?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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?

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

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

This activity investigates how you might make squares and pentominoes from Polydron.

How many different triangles can you make on a circular pegboard that has nine pegs?

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.

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

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

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

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 many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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.

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.

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....?

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

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