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?

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

How many trapeziums, of various sizes, are hidden in this picture?

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?

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?

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

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.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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

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

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?

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

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.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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?

How many models can you find which obey these rules?

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'.

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

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

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

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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?

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

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. . . .

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

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?

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

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?

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?

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

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?

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

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

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?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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?

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

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

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?