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, ...

I understand that -- but it seems like your solution is the same as Charlie's -- the only thing you left out is the program, which he provided for you. I suppose Ady was just noting that for the benefit of other readers.