1. What is the meaning of this sequence?
*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:

**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

[1,2,3,2,1,2,3,4,2,1,2,3,4,3,2,3,4,5,3,2,....]2. Take an integer

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=4Therefore, 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.

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.

CategoryMath

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