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?

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

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.

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.

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.

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.

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

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.

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

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.

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

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

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?”

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

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?

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?

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

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

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.

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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

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

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

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?

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

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. . . .

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

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

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

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?

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.

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

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.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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. . . .

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?

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

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

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

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.