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

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

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

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

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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....?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

How many different shapes can you make by putting four right- angled isosceles triangles together?

In this matching game, you have to decide how long different events take.

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?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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 fill in the empty boxes in the grid with the right shape and colour?

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?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Try out the lottery that is played in a far-away land. What is the chance of winning?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

What happens when you try and fit the triomino pieces into these two grids?

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

How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

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?

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.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

My coat has three buttons. How many ways can you find to do up all the buttons?