Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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 trapeziums, of various sizes, are hidden in this picture?

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

This activity investigates how you might make squares and pentominoes from Polydron.

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

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

How many different triangles can you make on a circular pegboard that has nine pegs?

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

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?

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

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?

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?

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?

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 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?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

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

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

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.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

What could the half time scores have been in these Olympic hockey matches?

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

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?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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?

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?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Can you find all the different triangles on these peg boards, and find their angles?

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

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.