Are these statements always true, sometimes true or never true?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

How many centimetres of rope will I need to make another mat just like the one I have here?

An investigation that gives you the opportunity to make and justify predictions.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Make some loops out of regular hexagons. What rules can you discover?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Here are two kinds of spirals for you to explore. What do you notice?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Are these statements always true, sometimes true or never true?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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.

Can you describe this route to infinity? Where will the arrows take you next?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

It starts quite simple but great opportunities for number discoveries and patterns!

Watch this animation. What do you see? Can you explain why this happens?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

What's the largest volume of box you can make from a square of paper?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

This challenge asks you to imagine a snake coiling on itself.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?