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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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?

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?

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?

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?

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 problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Investigate the different ways you could split up these rooms so that you have double the number.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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.

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

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?

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

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

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?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

How many models can you find which obey these rules?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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 investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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?

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

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.

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

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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?

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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.

Explore the triangles that can be made with seven sticks of the same length.

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?