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

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?

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

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

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?

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

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.

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.

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.

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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?

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

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.

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

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?

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.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

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

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

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

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

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

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

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

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

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

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

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Try out this number trick. What happens with different starting numbers? What do you notice?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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

This activity involves rounding four-digit numbers to the nearest thousand.

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

Got It game for an adult and child. How can you play so that you know you will always win?