Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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?

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?

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?

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?

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?

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.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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.

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.

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?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

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

The clues for this Sudoku are the product of the numbers in adjacent squares.

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

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

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

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 Sudoku that uses transformations as supporting clues.

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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.

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 based on clues that give the differences between adjacent cells.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

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

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

A Sudoku with clues given as sums of entries.

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.

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

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

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

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

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.

How many solutions can you find to this sum? Each of the different letters stands for a different number.