Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

A introduction to how patterns can be deceiving, and what is and is not a proof.

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

Can you find a rule which relates triangular numbers to square numbers?

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which connects consecutive triangular numbers?

Use Farey sequences to obtain rational approximations to irrational numbers.

Find the smallest value for which a particular sequence is greater than a googol.

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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?

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

This article for teachers describes the exchanges on an email talk list about ideas for an investigation which has the sum of the squares as its solution.

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

This resource contains interactive problems to support work on number sequences at Key Stage 4.

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.

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Join in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.

Can you puzzle out what sequences these Logo programs will give? Then write your own Logo programs to generate sequences.

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

A story for students about adding powers of integers - with a festive twist.

Formulate and investigate a simple mathematical model for the design of a table mat.

Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.