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

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

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?

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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?

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

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

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

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....?

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.

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?

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

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?

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 put the numbers 1 to 8 into the circles so that the four calculations are correct?

A Sudoku with clues given as sums of entries.

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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

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?

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.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

Can you find all the different ways of lining up these Cuisenaire rods?

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

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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

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

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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

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

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.

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

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

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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

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?

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?