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?

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?

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

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?

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

What happens when you try and fit the triomino pieces into these two grids?

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

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?

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

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?

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 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?

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?

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?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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?

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

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

How many models can you find which obey these rules?

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?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as 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?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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

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!

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

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.

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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.

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

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!

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

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.