1. What is the meaning of this sequence?
  [1,2,3,2,1,2,3,4,2,1,2,3,4,3,2,3,4,5,3,2,....]

2. Take an integer ''x'' greater than zero. Add together the squares of each digit. Repeat. ''x'' is a "happy number" if the result ever becomes 1. For example:
  11: 1+1=2
  2: 4=4
  4: 16=16
  16: 1+36=37
  37: 9+49=58
  58: 25+64=89
  89: 64+81=145
  145: 1+16+25=42
  42: 16+4=20
  20: 4+0=4
Therefore, 11 is not a happy number in base ten. It can be proved that all numbers that are not happy numbers in base ten make the same sequence repeating: 4,16,37,58,89,145,42,20,4,... Prove that in base 4, all numbers are happy numbers.

----

'''Answers'''

1. Ask the Romans! ''(Actually, you are close to the right answer! But it is not completely correct, and you have to be more specific as well.)'' [I could have been specific, but the clue proves I know without totally giving the game away.] ''(Oh, OK, you know, you win!)''

2. Trial and error. ''(No. You have to do it without trial and error. Trial and error is not proving anything!)'' [Plus induction, obviously. Someone else has completed the proof below.]

----

2.  Sketch: 1, 2, and 3 digit numbers are happy (Try them all; 1 itself is trivially happy, despite being the loneliest number). For an n digit number with n>3, we have 9n<4**(n-1), and the sum of squares of digits is at most 9n and so has fewer than n digits, and is happy by ProofByInduction. ''(OK, I guess this is correct.)''

----

''(I am AaronBlack and I made this page. Somebody else invented HappyNumber''''''s in base ten, and I think I invented HappyNumber''''''s in bases other then base ten.)''

See also: http://mathworld.wolfram.com/HappyNumber.html
----
CategoryMath