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?

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?

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?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

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

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

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.

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?

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.

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

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.

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.

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?

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

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?

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

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

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

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?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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?

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?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

How many models can you find which obey these rules?

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

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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

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

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

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

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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

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.

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

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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