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?

How many models can you find which obey these rules?

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

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

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?

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

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.

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

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?

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.

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

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.

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?

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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

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 you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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.

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?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

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 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?

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

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

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

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

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

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

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?

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

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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?

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?

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?

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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?

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

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

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

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