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, ...

Actually it was Charlie that did the checking. I merely outlined the proof without doing the number crunching. I did not prove there were two cycle only, only that there must be cycles. Ady is correct to point out that leaves room for more cycles.

The subject line of my post was meant to imply that it was proof enough for me, even though I did not fully prove the problem.