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?

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?

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

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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. . . .

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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

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. . . .

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

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

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.

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?

Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

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?

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

"Tell me the next two numbers in each of these seven minor spells", chanted the Mathemagician, "And the great spell will crumble away!" Can you help Anna and David break the spell?

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

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

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

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

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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.

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

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

An environment which simulates working with Cuisenaire rods.

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

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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

Investigate these hexagons drawn from different sized equilateral triangles.

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.

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

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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