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

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

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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.

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?

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

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 particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

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

This Sudoku, based on differences. Using the one clue number can you find the solution?

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?

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.

Four small numbers give the clue to the contents of the four surrounding cells.

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.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

A few extra challenges set by some young NRICH members.

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

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

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?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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.

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

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

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

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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

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

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

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.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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