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 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...

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

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?

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?

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

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 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. . . .

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. . . .

Can all unit fractions be written as the sum of two unit fractions?

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?”

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?

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.

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?

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

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?

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?

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.

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

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?

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

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

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

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

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.

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. . . .

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

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

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

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

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

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Got It game for an adult and child. How can you play so that you know you will always win?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

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

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

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?

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

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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?

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

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