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?

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

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.

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 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.

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?

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

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?

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

How many models can you find which obey these rules?

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

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten 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.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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?

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?

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

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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?

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?

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

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

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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

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. . . .

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

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

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?

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

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?

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

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

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?

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

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 clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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

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.