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.

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

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.

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

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.

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?

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?

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?

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.

How many models can you find which obey these rules?

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

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

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?

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

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.

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.

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?

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?

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?

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?

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

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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

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

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.

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?

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?

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?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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. . . .

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?

If you had 36 cubes, what different cuboids could you make?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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?

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.

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?

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

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