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?

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?

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.

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.

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?

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?

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

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

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

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?

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 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?

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?

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?

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?

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

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

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?

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?

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

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

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?

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

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

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

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.

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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?

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

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

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

In how many ways can you stack these rods, following the rules?

How many models can you find which obey these rules?

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

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....?

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?

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.

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

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

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

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

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.