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?

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

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

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?

Investigate these hexagons drawn from different sized equilateral triangles.

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.

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?

How many different sets of numbers with at least four members can you find in the numbers in this box?

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?

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Investigate what happens when you add house numbers along a street in different ways.

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

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

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

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.

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 ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

An environment which simulates working with Cuisenaire rods.

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

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

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?

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

"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?

July 1st 2001 was on a Sunday. July 1st 2002 was on a Monday. When did July 1st fall on a Monday again?

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

Liitle Millennium Man was born on Saturday 1st January 2000 and he will retire on the first Saturday 1st January that occurs after his 60th birthday. How old will he be when he retires?

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

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

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

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?

Investigate the successive areas of light blue in these diagrams.

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?

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.

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

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

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?

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?

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.

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?

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

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

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

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