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

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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.

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

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

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

This activity investigates how you might make squares and pentominoes from Polydron.

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

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?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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?

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

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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.

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

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Can you draw a square in which the perimeter is numerically equal to the area?

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?

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?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

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

How many trains can you make which are the same length as Matt's, using rods that are identical?

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

How many models can you find which obey these rules?

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.

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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?

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. . . .

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

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

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?

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?

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

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

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

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

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

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

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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