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

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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

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

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

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?

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.

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

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

What happens when you try and fit the triomino pieces into these two grids?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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

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?

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

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?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of these convex shapes?

Can you fit the tangram pieces into the outline of the rocket?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Can you work out what is wrong with the cogs on a UK 2 pound coin?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

Can you fit the tangram pieces into the outline of the telescope and microscope?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

Can you fit the tangram pieces into the outline of this plaque design?

Can you fit the tangram pieces into the outline of this goat and giraffe?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

Can you fit the tangram pieces into the outlines of the workmen?

Can you fit the tangram pieces into the outlines of the candle and sundial?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Here are shadows of some 3D shapes. What shapes could have made them?

Which of these dice are right-handed and which are left-handed?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?