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

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?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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?

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.

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?

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

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

Explore the effect of reflecting in two intersecting mirror lines.

Explore the effect of reflecting in two parallel mirror lines.

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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

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

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

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

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?

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

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.

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?

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

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?

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.

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?”

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.

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.

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

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.

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

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 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.

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.

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?

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. . . .

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

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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

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

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

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?

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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

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.