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

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?

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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.

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

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?

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

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

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.

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

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

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.

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

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

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?

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

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

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

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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?

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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?

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?

Can you find the values at the vertices when you know the values on the edges?

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?”

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

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?

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.

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

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

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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

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

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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