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?

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?

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?

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

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?

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?

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 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.

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

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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?

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?

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?

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

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 different triangles can you draw on the dotty grid which each have one dot in the middle?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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

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. . . .

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

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?

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.

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 rhythms can you make by putting two drums on the wheel?

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?

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

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

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

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?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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.

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

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

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

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....?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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

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

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.

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