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?

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?

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

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

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

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?

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?

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?

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

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?

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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

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.

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

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

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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?

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

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?

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

How many models can you find which obey these rules?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

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

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

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.

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

I cut this square into two different shapes. What can you say about the relationship between them?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

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?

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?

An investigation that gives you the opportunity to make and justify predictions.

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!

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

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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?

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.

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?

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?

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?

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?