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?

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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

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

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

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

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

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.

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

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

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.

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

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?

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

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

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

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?

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

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?

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?

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

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

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

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

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?

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?

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

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

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

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?

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

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

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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?