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. . . .

Three frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

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 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. . . .

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.

Which of these pocket money systems would you rather have?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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

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

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Make some loops out of regular hexagons. What rules can you discover?

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

There are lots of ideas to explore in these sequences of ordered fractions.

Can you find a rule which connects consecutive triangular numbers?

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

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

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.

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.

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

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

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

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

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

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

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?

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

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?

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.

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

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. . . .

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.

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

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

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

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.

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and. . . .

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. . . .

A tower of squares is built inside a right angled isosceles triangle. What fraction of the area of the triangle is covered by the squares?

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

What's the greatest number of sides a polygon on a dotty grid could have?

Can you find a way to identify times tables after they have been shifted up?