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?

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?

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?

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?

What is the best way to shunt these carriages so that each train can continue its journey?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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?

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?

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?

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.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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

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?

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

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

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

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?

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?

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.

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?

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?

How many models can you find which obey these rules?

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

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

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?

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.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

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

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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

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

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?

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.

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

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

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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?

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

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 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?

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

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

This task follows on from Build it Up and takes the ideas into three dimensions!

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

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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