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?

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?

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?

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?

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

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?

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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

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?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

How many models can you find which obey these rules?

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?

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?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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?

Find out what a "fault-free" rectangle is and try to make some of your own.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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

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

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!

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?

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

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?

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?

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

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

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

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.

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?

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

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

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?

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?

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!

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.

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.

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