Consider a series of numbers, defined as follows:
Starting with any natural number, each member is a sum of the squares of the previous member`s digits.

Prove : The series always reaches either a stuck-on-one sequence: 1,1,1… or a closed loop of the following 8 numbers: 145,42,20,4,16,37,58,89, ...

Ex1: 12345,55,50,25,29,85,89,145….. etc
Ex2: 66,72,53,34,25,29,85,89,145…
Ex3: 91,10,1,1,1…..

(In reply to solution enough by Jer)

Whatever you said is true, but you did not provr that there only two possibilities : a  loop of 1,1.. and  the closed loop of 8  specific numbers.

Posted by Ady TZIDON on 2011-03-07 17:57:23