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 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.

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?

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

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

How many models can you find which obey these rules?

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 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.

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?

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

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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

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

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

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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?

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

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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?

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

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

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

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

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

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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?

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

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

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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.

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?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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?

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?

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

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.

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

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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. . . .

How many possible necklaces can you find? And how do you know you've found them all?