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?

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

How many models can you find which obey these rules?

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

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

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

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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?

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

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?

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

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?

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?

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?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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.

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

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?

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

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 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?

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.

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

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?

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.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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

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.

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. . . .

This activity investigates how you might make squares and pentominoes from Polydron.

In how many ways can you stack these rods, following the rules?

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.

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

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

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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

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?

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.

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?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

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

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.

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?