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. . . .

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Can you find the area of a parallelogram defined by two vectors?

An article which gives an account of some properties of magic squares.

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

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

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?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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?

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?”

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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

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

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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?

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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.

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.

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

A game for 2 players with similaritlies 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.

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?

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.

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?

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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.

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 loser is the player who takes the last counter.

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

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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.

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

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

An account of some magic squares and their properties and and how to construct them for yourself.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

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

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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

To avoid losing think of another very well known game where the patterns of play are similar.

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