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

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

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 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?

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?

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.

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.

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

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

A Sudoku with clues given as sums of entries.

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?

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?

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

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

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

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

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

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

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

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!

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?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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.

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

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

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

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

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

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?

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.

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?

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?

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

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.

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

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?

A Sudoku that uses transformations as supporting clues.

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.

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.

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 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!

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

Find out about Magic Squares in this article written for students. Why are they magic?!

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.