Use the clues about the symmetrical properties of these letters to place them on the grid.

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

How many trapeziums, of various sizes, are hidden in this picture?

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

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

Can you fill in the empty boxes in the grid with the right shape and colour?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

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

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?

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?

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

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?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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?

Can you find all the different triangles on these peg boards, and find their angles?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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?

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. . . .

Find all the numbers that can be made by adding the dots on two dice.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

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

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?

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

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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

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?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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?

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