How many models can you find which obey these rules?

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

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?

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?

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?

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?

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?

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?

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?

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?

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 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?

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

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?

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

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.

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?

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.

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'.

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

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?

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?

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

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

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

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

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?

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.

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 have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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.

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

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.

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!

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

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

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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.

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