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

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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.

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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

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.

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?

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

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.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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

Can you figure out how sequences of beach huts are generated?

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

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Can you find sets of sloping lines that enclose a square?

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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.

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

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

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?

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

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

It would be nice to have a strategy for disentangling any tangled ropes...

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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.

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?”