Happy Numbers

1
Happy Numbers

• By Kylie Chaffer and Erin Holliday

2
Introduction

• Think of a number
• Square each of its digits
• Add the squares to get a second number

3
How do you reach a happy number?

• If the sequence reaches 1, then the original
  number is called happy.
• If not, it is called sad.

4
Example

• 2 3
• 2² 3²
• 4 9 1 3
• 1² 3²
• 1 9 1 0
• 1² 0²
• 1 0 1

5
Which aspects did we investigate?

• Were there any visible patterns in the sequences?
• How many terms were in each sequence before the
  cycle was reached?
• What proportion of the numbers from 1 to 100 are
  happy/sad? Etc
• Are happy numbers more often odd or even?
• What happens if we cube the numbers rather than
  square them?

6
PATTERNS

• If the sequence reaches 58 or 85, it would
  eventually repeat the cycle. E.g.
• 26, 40, 16, 37, 58, 89, 145, 42, 20, 4
• 8, 64, 52, 29, 85, 89, 145, 42, 20, 4, 16, 37,
  58

7
PATTERNS

• There are 3 different types of sequences
• 4-loop
• 4, 16, 37, 58, 89
• 5-loop
• 5, 25, 29, 85, 89
• 1-loop
• 1, 1

8
PATTERNS

• When a happy number was found we added the digits
  of the number to see if a pattern formed.
• E.g.
• 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68,
• (1) (7) (1) (4) (10) (5) (10)
  (4) (5) (8) (13) (14)
• 70, 79, 82, 86, 91, 94, 97, 100
• (7) (16) (10) (14) (10) (13)
  (16) (1)

9
Happy numbers
No multiples of 3 or 6 below one hundred are
happy numbers. Every number with a 5 in it is a
sad number.
Multiples of three
What happens after 100?
10
500s 5000s

• When looking at these numbers we did not always
  complete the entire sequence, if we knew how it
  would eventuate.
• E.g. 532, 38
• First happy number in the 500s was 536.
• First happy number in the 5000s was 5111.
• Therefore there were happy numbers that had a 5
  in it (536, 5175, 5257 etc.), disproving our
  theory. There were also numbers ending in 5 that
  were happy numbersanother dead end.

11
After 100

• We continued the table until we reached 1500 and
  found
• - no multiples of 3 until 129
• - no multiples of 6 until 192
• - no numbers with 5 in it until 356
• - no numbers ending in 5 until 365
• CONCLUSION There are no obvious patterns
  between happy numbers that help us to easily
  determine which numbers are happy and which
  numbers are sad.

12
TERMS

• Each sequence contained a certain number of terms
  before reaching 58, 85 or 1.
• E.g. 39, 90, 81, 65, 61, 37, 58 (7 terms)
• 17, 50, 25, 29, 85 (5 terms)
• 28, 68, 100, 1 (4 terms)
• We still found no pattern

13
PROPORTIONS

• What proportion of the numbers from 1 to 100 are
  happy/sad? Etc
• 19 / 100
• 32 / 200
• 44 / 300
• 66 / 400
• 76 / 500

1-10 3 11-20 2 21-30 2 31-40 2 41-50
2 51-60 0 61-70 2 71-80 1 81-90 1 91-100
4
11/50 19/100 (8) 26/150 (7) 32/200 (6) 39/250
(7) 44/300 (5) 54/350 (10) 66/400 (12) 70/450
(4) 76/500 (6)
14
ODDS AND EVENS
Out of the first 100 happy numbers, 50 of these
were odd.
15
CUBED NUMBERS
What happens if we cube the numbers rather than
square them?

• 3³ 2 7
• 2³ 7³
• 8 343 3 5 1
• 3³ 5³ 1³
• 27 125 1 1 5 3

16
CUBED NUMBERS

• As seen in the example above, most sequences
  ended when the following number was made up of
  the same digits.
• E.g. 351 is the same as 153 or 531
• The only happy cubed numbers found were 1, 10,
  100, 1000 etc.

17
HAPPY NUMBERS

• 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68,
• 70, 79, 82, 86, 91, 94, 97, 100, 103, 109,
• 129, 130, 133, 139, 167, 176, 188, 190, 192
• 193, 203, 208, 219, 226, 230, 236, 239,
• 262, 263, 280, 291, 293, 301, 302, 310,
• 313, 319, 320, 329, 331, 338, 356, 362,
• 365, 376, 379, 383, 386, 391, 392, 397,
• 404, 409, 440, 446, 464, 469, 478, 487,
• 490, 496, 536, 556, 563, 565, 566

18
E-FOLIO

• Please note that we have inserted this
  investigation into our e-portfolios, under
  Curriculum and Knowledge.