Math Quiz Six

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,....]

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:
1. : 1+1=2
2. : 4=4
3. : 16=16
4. : 1+36=37
5. : 9+49=58
6. : 25+64=89
7. : 64+81=145
8. : 1+16+25=42
9. : 16+4=20
10. : 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

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!)

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.]

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 HappyNumbers in base ten, and I think I invented HappyNumbers in bases other then base ten.)

CategoryMath