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Broad Topics > Numbers and the Number System > Patterned numbers

### Tower of Hanoi

##### Stage: 4 Challenge Level:

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

### Caterpillars

##### Stage: 1 Challenge Level:

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

### Harmonic Triangle

##### Stage: 3 Challenge Level:

Can you see how to build a harmonic triangle? Can you work out the next two rows?

### How Many Miles to Go?

##### Stage: 3 Challenge Level:

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

### Sixty-seven Squared

##### Stage: 5 Challenge Level:

Evaluate these powers of 67. What do you notice? Can you convince someone what the answer would be to (a million sixes followed by a 7) squared?

### Sept 03

##### Stage: 3 Challenge Level:

What is the last digit of the number 1 / 5^903 ?

### How Old Am I?

##### Stage: 4 Challenge Level:

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

### Odd Differences

##### Stage: 4 Challenge Level:

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.

### Domino Sets

##### Stage: 2 Challenge Level:

How do you know if your set of dominoes is complete?

### Rods and Rods

##### Stage: 2 Challenge Level:

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

### Unlocking the Case

##### Stage: 2 Challenge Level:

A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?

### On the Importance of Pedantry

##### Stage: 3, 4 and 5

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

### Four Coloured Lights

##### Stage: 3 Challenge Level:

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

### Sorting the Numbers

##### Stage: 1 and 2 Challenge Level:

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

### The Numbers Give the Design

##### Stage: 2 Challenge Level:

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

### Birds in the Garden

##### Stage: 1 and 2 Challenge Level:

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

### Sorting Numbers

##### Stage: 1 Challenge Level:

Use the interactivity to sort these numbers into sets. Can you give each set a name?

### Investigating Pascal's Triangle

##### Stage: 2 and 3 Challenge Level:

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

### When Will You Pay Me? Say the Bells of Old Bailey

##### Stage: 3 Challenge Level:

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

### You Owe Me Five Farthings, Say the Bells of St Martin's

##### Stage: 3 Challenge Level:

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

### Oranges and Lemons, Say the Bells of St Clement's

##### Stage: 3 Challenge Level:

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

### Tables Without Tens

##### Stage: 2 Challenge Level:

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

### Colour Building

##### Stage: 3 Challenge Level:

Using only the red and white rods, how many different ways are there to make up the other colours of rod?

### Generating Number Patterns: an Email Conversation

##### Stage: 2, 3 and 4

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.

### Shedding Some Light

##### Stage: 2 Challenge Level:

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

### Hundred Square

##### Stage: 1 Challenge Level:

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

### All Seated

##### Stage: 2 Challenge Level:

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

### Pattern Power

##### Stage: 1, 2 and 3

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

### Squares, Squares and More Squares

##### Stage: 3 Challenge Level:

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

### Taking Steps

##### Stage: 2 Challenge Level:

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

### Lastly - Well

##### Stage: 3 Challenge Level:

What are the last two digits of 2^(2^2003)?

### A One in Seven Chance

##### Stage: 3 Challenge Level:

What is the remainder when 2^{164}is divided by 7?

### Rolling Coins

##### Stage: 4 Challenge Level:

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

### Top-heavy Pyramids

##### Stage: 3 Challenge Level:

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

### Sept03 Sept03 Sept03

##### Stage: 2 Challenge Level:

This number has 903 digits. What is the sum of all 903 digits?

### Hidden Squares

##### Stage: 3 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

### Counting Binary Ops

##### Stage: 4 Challenge Level:

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.

### Magic Squares II

##### Stage: 4 and 5

An article which gives an account of some properties of magic squares.

### Magic Squares

##### Stage: 4 and 5

An account of some magic squares and their properties and and how to construct them for yourself.

### Try to Win

##### Stage: 5

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

### Whole Number Dynamics V

##### Stage: 4 and 5

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

### Magical Maze - 35 Activities

##### Stage: 4 and 5

Investigations and activities for you to enjoy on pattern in nature.

### Whole Number Dynamics IV

##### Stage: 4 and 5

Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?

### Whole Number Dynamics III

##### Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

### Whole Number Dynamics II

##### Stage: 4 and 5

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

### Whole Number Dynamics I

##### Stage: 4 and 5

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Transformation Tease

##### Stage: 2 Challenge Level:

What are the coordinates of this shape after it has been transformed in the ways described? Compare these with the original coordinates. What do you notice about the numbers?

### Back to Basics

##### Stage: 4 Challenge Level:

Find b where 3723(base 10) = 123(base b).

### Power Crazy

##### Stage: 3 Challenge Level:

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?