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?

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?

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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

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

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.

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

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

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?

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

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

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

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

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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.

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.

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?

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?

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?

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

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

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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

Can you find a way of counting the spheres in these arrangements?

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

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

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

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

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?

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

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?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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.

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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

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.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?