Happy numbers

1
Happy numbers

• What is a Happy number.
• Pick a number
• Square each of the digits
• Then add both of the answers together
• Continue to do this sequence until you and up
  with the number 1.

2
Example of Happy Numbers

• 23
• (2x2) (3x3)
• 4 9 13
• (1x1) (3x3)
• 1 9 10
• (1x1) (0x0)
• 1
  0 1
• Therefore 23 equals a happy number and
• So does 13 and 10.

3

• All Happy numbers sequence do not have to be
  finished, you can just assume that the number
  will finish the same when it copies the same
  number in a sequence before.
• 7, (7x7)49, (4x4)(9x9)97, (9x9)(7x7)130,
• (1x1)(3x3)(0x0)10, (1x1)(0x0)1
• 94, (9x9)(4x4)97, (9x9)(7x7)130,
• (1x1)(3x3)(0x0)10, (1x1)(0x0)1
• As soon above the sequence continues at 97

4

• Through my investigations Sad numbers can also be
  worked out in the same way and they all end in
  same digit.
• 2, (2x2)4, (4x4)16, (1x1)(6x6) 37,
  (3x3)(7x7)58,
• (5x5)(8x8)89
• (8x8)(9x9)145, (1x1)(4x4)(5x5)42,
  (4x4)(2x2)20,
• (2x2)(0x0) 4
• 5, (5x5) 25, (2x2)(5x5)29, (2x2)(9x9)85,
  (8x8)(5x5)89,
• (8x8)(9x9)145, (1x1)(4x4)(5x5)42,
  (4x4)(2x2)20,
• (2x2)(0x0) 4

5
There are many aspects of my investigations with
Happy numbers and with these results it would be
safe to say that either when you times a number
by ten or mix around numbers the results will end
up being the same as before.

• 1, 10, 100, 1000, 10 000, 100 000, 1 000 000
• (1x1) 1or (1x1)(0x0)(0x0)(0x0)(0x0)(0x0) 1
• 13, 31, 103, 301, 310, 130, 1003, 1030, 1300,
  3100, 3010
• 13, (1x1)(3x3)10, (1x1)(0x0)1
• 1300, (1x1)(3x3)(0x0)(0x0)10, (1x1)(0x0)1
• This investigation can also work on Sad numbers.

6
Happy Sad numbers20 Happy numbers
12 happy numbers
7
Continue

• From the tables above there where several
  patterns which I tried to develop,
• In each hundred there is a middle point and I
  tried to find if they where even on each side.
• 1,7,10,13,19,23,28,31,32,44,49,/
  68,70,79,82,86,91,94,97,100
• 1-100 the Happy numbers do not show equal amount
  on either side of fifty, under fifty there are 11
  and over fifty there are 9, there is 2 more under
  fifty. In this case the pattern would be 2 under
  more happy numbers under fifty but the pattern
  does not continue, because in the next 101-200
  there are 6 on either side of fifty and that
  pattern also doesnt continue with one being 2
  less and the next being equal.

8
When investigating patterns I use a website
www.wschnei.de/digit-related-numbers/happy-numbers
-list.html which showed all the Happy numbers
from 1-10 000, this allowed me to continue to see
if a pattern would appear from the amount of
happy numbers in each hundreds.

• 1-100 20 801-900 13 1601-1700 20
• 101-200 12 901-1000 19 1701-1800 19
• 201-300 12 1001-1100 11 1801-1900 11
• 301-400 22 1101-1200 16 1901-2000 19
• 401-500 10 1201-1300 19 2001-2100 12
• 501-600 5 1301-1400 14 2101-2200 13
• 601-700 19 1401-1500 13
• 701-800 11 1501-1600 7
• I thought at the beginning I might be able to
  find a pattern, with the first having the most,
  second has a middle amount and third has the
  least but that didnt work. You can see that
  there isnt any pattern with the numbers or is
  there a number around the same number. The
  average number that each hundred has is
  14.40909091

9
Cubing the digits, follow the same basic rule.

• These digits have a pattern that I have found in
  a small investigation.
• 3, (3x3x3)27, (2x2x2)(7x7x7)351,
  (3x3x3)(5x5x5)(1x1x1)153, (1x1x1)(5x5x5)(3x3x
  3)153
• 8, 512, 12518134, 1276492, 7298737,
  34327343713, 343127371, 273431371
• 7, 343, 276427118, 11512514, 125164190,
  17290730, 343270370, 273430370
• 9, 729, 34387291080, 105120513,
  125127153, 112527153
• The end numbers will tell the ending of the
  sequence, when your answer has the same digits as
  the answer before the end result will be that the
  answer will, in end stay the same number.

10
On a website I found an interesting figure, when
you cube. At www.articlesforeducators.com/articles
.asp?aid1

• 1 125 27
• 153, (1x1x1)(5x5x5)(3x3x3)153
• The only other number that does the same is 1
• 1, (1x1x1)1
• Therefore these numbers are not the same as the
  slide before hand.

11
Conclusion

• In conclusion I have found no large pattern
  that relates to Happy or Sad numbers so it can be
  easy to predict the next number. Although there
  are many little patterns in Happy and Sad numbers
  such as all Happy numbers end with a one and all
  Sad numbers end with a four, all the sequences
  can relate to another sequence so therefore you
  will not have to finish the entire sequence to
  find out the answer. There are other patterns
  that I have searched for without having any luck.
  I also took a quick look at happy cube numbers
  and found there is also a pattern that can be
  found in the last two answers so you can predict
  an ending. Happy numbers seem to be unpredictable
  or further research should be held.