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?

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

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. . . .

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

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

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

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

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.

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?

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

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 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. . . .

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

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

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

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?

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?

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

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

Investigate the successive areas of light blue in these diagrams.

An environment which simulates working with Cuisenaire rods.

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.

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

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

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

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

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

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

"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?

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

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

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.

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?

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

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

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

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?

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

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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?

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.

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

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