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

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.

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

How many models can you find which obey these rules?

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?

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?

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.

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.

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?

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

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

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?

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.

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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?

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

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

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

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?

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

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?

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

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

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

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?

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?

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?

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

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

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

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

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?

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?

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

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

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

An investigation that gives you the opportunity to make and justify predictions.

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

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

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

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

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

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

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?

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. . . .

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