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

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

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

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?

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

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.

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?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Can you draw a square in which the perimeter is numerically equal to the area?

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?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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?

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?

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.

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

Find the values of the nine letters in the sum: FOOT + BALL = GAME

Use the differences to find the solution to this Sudoku.

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?

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

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?

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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?

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. . . .

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

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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

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

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

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

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

A few extra challenges set by some young NRICH members.

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.

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?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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?

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

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

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

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.

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.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Given the products of adjacent cells, can you complete this Sudoku?

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

Can you use the information to find out which cards I have used?

These practical challenges are all about making a 'tray' and covering it with paper.

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