How many different symmetrical shapes can you make by shading triangles or squares?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

A challenging activity focusing on finding all possible ways of stacking rods.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

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

Use the clues about the symmetrical properties of these letters to place them on the grid.

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

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.

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.

Given the products of diagonally opposite cells - can you complete this Sudoku?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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

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?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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!

A Sudoku that uses transformations as supporting clues.

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 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!

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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?

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

How many models can you find which obey these rules?

Two sudokus in one. Challenge yourself to make the necessary connections.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you find all the ways to get 15 at the top of this triangle of numbers?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?