Happy Numbers

Introduction

We will continue to work with square numbers today! Look at the sequence of square numbers on the right, how many more square numbers can you write in 4 minutes?

Ready? Go!

Investigating Happy Numbers

Some numbers are happy, some are sad! To find out whether a number is happy, follow these steps:

  1. Square each of the number's digits

  2. Add the answers and repeat.

  3. If you end up with 1 the number is happy!

Look at the number on the right as an example, if you are still unsure, take a look at this video on Numberphile.

Now show whether the following numbers are happy or not, make sure you show the steps of your working. Can you present these as chains? For the sad numbers above, what have you noticed?

  • 23

  • 7

  • 4

  • The floor you live on

  • Your street address

  • Your telephone number

  • Your parents' car license plate number (for a challenge, figure this out in your head!)

Now, what if you are asked to find out the percentage of numbers from 0-99 that are happy or not?


Support

if you need some prompts, take a look at these:

  • What is a systematic way of working this out?

  • How can you represent it on a diagram?

    • Hint: think about chains and loops.

  • Can you think of any shortcut to reduce working?

  • How can you use the 0-99 grid (PRINT ME) on the right to record findings?

    • Hint: think about colouring happy and sad numbers in two different colours.

  • What patterns do you notice from your coloured grid?

If you wish to, you may use this to help check whether a number is happy or not!

Further Questions and Challenges

  • What is the longest chain you can find?

  • What about the longest loop?


  • Which number, in the 30’s, leads immediately to a number one less than itself?

  • Find two consecutive happy numbers.

  • There are two happy numbers between 11 and 20 and two happy numbers between 21 and 30. Does this sequence continue? If not, is there a different rule we can use?

  • What is the longest run on consecutive happy numbers?

  • Are there any other “dead ends” other than 1?


  • Is there a way to go backwards?

    • Try working backwards from say 8. The number before that is 22, can you explain why?

    • What are the number before 36? There are two, can you explain why?

    • Can you find some strategies for how to work backwards from 1 for happy numbers?


  • Is there an infinite number of happy numbers?

  • Is there an infinite number of unhappy numbers?

  • Does the ratio of happy to unhappy numbers approach a limit?

  • Can you find any patterns in the sequence of happy numbers?

  • Ellen told me that every number comes down to either one or four- is this true? Why/ why not?


  • 153 called a Happy Cube. Similar to the happy number process, only cubing instead of squaring.

    • 153: 13 + 53 + 33 = 1 + 125 + 27 = 153

    • There are other happy cubes; can you find any?

  • 4,210,818 is a happy seventh-power.

    • 47 + 27 + 17 + 07 + 87 + 17 + 87 = 4,210,818

    • Can you find any other happy other-power numbers?

Further Practice

The essential skill introduced in this lesson is "square numbers". Other skills you may also wish to recap/ learn about are "square root", "cube numbers" and "cube root". Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.


Transum

Here is a game to practise recognising square numbers.

Try this self-marked exercise on transum to check your understanding of roots.

Looking for something fun to play with your friend? Try this "Square Pairs" game. Where you take turns to select two numbers that add up to a square number. The winner is the last person standing. For a similar activity to do own your own try this.

Extension

Take a look at number gossip. Put a random number in and it will tell you about some of the number's properties. For example, 149 is an "evil" "twin", find out what that means!


nRich

For the coders among you, you may wish to learn how to use a spreadsheet to help find happy numbers. Take a look at this investigation for instructions and examples.

We maybe able to count on our 10 fingers, but what about Octopuses? Happy Octopus investigates happy numbers where all numbers are written in base 8.


If you think Happy Numbers are useless, take a look at how it helped Dr Who to escape a sticky situation!