Challenge Level

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

Challenge Level

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Challenge Level

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

Challenge Level

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

Challenge Level

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

Challenge Level

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

Challenge Level

What do these two triangles have in common? How are they related?

Challenge Level

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

Challenge Level

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Challenge Level

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Challenge Level

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Challenge Level

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Challenge Level

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

Challenge Level

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Challenge Level

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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?

Challenge Level

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?

Challenge Level

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

Challenge Level

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

Challenge Level

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

Challenge Level

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

Challenge Level

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?

Challenge Level

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

Challenge Level

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Challenge Level

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

Challenge Level

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Challenge Level

A follow-up activity to Tiles in the Garden.

Challenge Level

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

Challenge Level

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Challenge Level

Why does the tower look a different size in each of these pictures?

Challenge Level

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

Challenge Level

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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?

Challenge Level

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?

Challenge Level

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

Challenge Level

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Challenge Level

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?

Challenge Level

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?

Challenge Level

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?

Challenge Level

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?

Challenge Level

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

Challenge Level

What is the largest cuboid you can wrap in an A3 sheet of paper?