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?

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.

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.

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?

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?

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

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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

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

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?

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

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?

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

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

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

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?

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

How many models can you find which obey these rules?

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 cuboid that you can put in this box so that you cannot fit another that's the same into it?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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

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

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

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

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?

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?

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

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?

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

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.

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

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 shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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.

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?