Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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?

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?

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

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?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

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

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

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

How many trapeziums, of various sizes, are hidden in this picture?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

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?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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?

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.

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.

How many models can you find which obey these 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?

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

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. . . .

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal 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?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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?

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

A few extra challenges set by some young NRICH members.

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?