A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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.

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

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?

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

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

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

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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

What can you see? What do you notice? What questions can you ask?

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?

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?

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

Can you make a 3x3 cube with these shapes made from small cubes?

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?

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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. . . .

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?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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?

On which of these shapes can you trace a path along all of its edges, without going over any edge twice?

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?

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

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?

A cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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?

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?

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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?

Square It game for an adult and child. Can you come up with a way of always winning this game?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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

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?

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 loops of string have been used to make these patterns?

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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?