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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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.

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?

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

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

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

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?

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

Delight your friends with this cunning trick! Can you explain how it works?

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

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

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 work out how to win this game of Nim? Does it matter if you go first or second?

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?

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

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.

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.

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

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?

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

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

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?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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?

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. . . .

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

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

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?

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

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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

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.

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.

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?