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One Is the Loneliest Number But Also a Happy Number?

Consider the number 17...

12+72=50 .... 52+02=25 .... 22+52=29 .... 22+92=85 .... 82+52=89 .... 82+92=145 .... 12+42+52=42 .... 42+22=20 .... 22+02=4 .... 42=16 .... 12+62=37 .... 32+72=58 .... 82+52=89 ....

Any cyle under this process (e.g. 89...145...42......37...58) is called a Cheery Sequence. In the case of the number 17, its cycle has 8 terms.

Question 1: Do all positive integers generate Cheery Sequences?

Question 2: Is there a pattern in their periods...that is, do certain types of numbers produce certain cycles and/or periods?

New Information: A Happy Number is an integer whose Cheery Sequence has a period of 1. Why is 10 a Happy Number?

Question 3: Do all integers generate Cheery Sequences with periods of 8 or 1?

Question 4: Do all Happy Numbers have a Cheery Sequence or cycle involving only a repeating 1?

Question 5: Is there an infinite number of Happy Numbers?

Question 6: Are there a pair of consecutive Happy Numbers? Trio of consecutive Happy Numbers in a row? N-consecutive Happy numbers for any N?

Question 7: Does "Happiness" depend on the base? That is, investigate these same questions in bases other than base 10.


Hint: Just start playing...and answers for the questions will start to develop.


Solution Commentary: Hope your investigations were fun and fruitful, that is, you are Happy and Cheery as well!

If you want to explore the given relationships further, consider the "assumed" original source: Donald Duncan's "Happy Integers," Mathematics Teacher, November 1972, pp. 627-629...or even what may be the original source: A. Prges' "A Set With Eight Numbers," American Mathematical Monthly, 1945, pp. 379-383.