EMM111 Maths And Numeracy Assignment 2: Investigation Seminars'

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EMM111 Maths And Numeracy Assignment 2
Investigation Seminars.
HAPPY NUMBERS
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What Are Happy Numbers?

• A happy number is when you think of any number.
• Square each of the numbers digits and add the
  squares to get a second number.
• Repeat process number two on the second number,
  and add the squares to get a third number.
• Continue this process until a sequence is formed.
• For example 7s sequence is
• 7,49,97,130,10,1
• If a sequence reaches 1, then the number is
  HAPPY, if it doesnt the number is SAD.

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How to Calculate Happy and Sad Numbers.
(A Practical Example)

• Eg/ 7 77 49
• 4944 99 97
• 97 99 77 130
• 130 11 33 00 10
• 10 11 00 1
• 1 HAPPY NUMBER.

• Eg/ 4 44 26
• 16 11 66 37
• 37 33 77 58
• 58 55 88 89
• 89 88 99 145
• 145 11 44 55 42
• 42 44 22 20
• 20 22 00 4
• 4 Sad Number

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Processes of Investigating Happy
Numbers.

• The first weeks two weeks involved problem
  solving with a partner. During the two weeks all
  that are achieved was knowing what numbers are
  happy from one to thirty.
• I then consulted with Craig to see if a formula
  could be created to find Happy Numbers but after
  hours of trying a formula could not be reached.
• After this consultation it became clear that
  other alternatives would need to be used to find
  out what numbers are happy and sad.
• I began calculating the answers manually using a
  calculator but after two hours and twenty numbers
  later I decided that there would need to be a
  better alternative.
• I began researching on the internet and found the
  website
• www.wschnei.de/digit-related-numbers/
  happy-numbers-list.html
• This site was my life saver as it gave me all
  the happy numbers to one thousand. This allowed
  me to focus on how Happy Numbers occur rather
  then what are the happy numbers.

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The Results.
Here are the happy Numbers from 1-1000 over the
next two slides
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Happy Numbers Continued.
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What Does This Tell Us

• What Proportion of numbers are happy
• 1-20 11 happy numbers 55
• 1-100 20 happy numbers 20
• 1-1000 142 happy numbers 14.2

Conclusion 85.8 of numbers between 1 and 1000
are sad
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Predicting Numbers.

• Can you predict if any numbers will be happy or
  sad?
• Yes and NO
• Rules Associated with Calculating Happy Numbers
• Numbers with the same digits will provide the
  same results eg/ 319 931 are both happy
  numbers.
• The 0 digit has no impact or implication on the
  result.
• When calculating happy numbers all the numbers in
  the sequence leading up to one are happy numbers
  eg/ 7,49,130,10 1 are all happy numbers.
• For every happy number when you add a 0 the
  number also becomes happy eg/ 13 and 130 are both
  happy numbers.
• Myths about calculating happy numbers
• - All Happy Numbers are divisible by 7
• - Digits add up to 10

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Is There a Cycle

(how
do I know when to stop)

• During manual calculations I noticed a cycle
  developing that showed when a number was sad.
• There was a 8 number cycle that continually
  rotates and a number cant escape.
• The cycle is

58, 89, 145, 42, 20, 4, 16 37
The numbers keep repeating and the number can
never become happy.
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Investigation Summary.

• The Processes that I undertook to complete the
  Investigation the questions that I tried to
  answer were
• What is a Happy Sad Number?
• How do I calculate the numbers?
• What are the Happy Numbers?
• Does the Percentage of Happy numbers increase or
  decrease as we investigate more numbers?
• Are there any rules that help us calculate Happy
  numbers?
• Can we Predict Happy Numbers?
• How do we know when a number has no chance of
  becoming Happy?
• What is this cycle?
• Provide a table of Happy Numbers to see if any
  one else can see a pattern.

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Conclusions.

• The majority of numbers are sad.
• I couldnt find a formula to solve the
• investigation or tell us what numbers are
• happy or sad.
• You can predict happy numbers by knowing the
  results certain digits grouped together produce.
• Sad numbers produce a 8 number cycle so when you
  reach a number twice you know it is sad.
• Numbers with the same digits produce the same
  results.
• Giving a class full of students this
  investigation could keep them intrigued and
  mentally busy for hours.

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Evidence of E-Portfolio Inclusion

• To see this PowerPoint presentation on my
  E-Portfolio it can be seen on the website
• http//athene.riv.csu.edu.au/btoole02/