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, ...

No 3 (or more) digit number can ever be followed by a larger term. This means if the starting number has more than 3 digits, some successive term will have fewer than 3 digits.

If you then continue finding terms there must eventually be a repeat and you have a cycle.

I suppose this doesn't prove you get one of the two cycles given in the problem. I suppose if I had the patience to check where all the numbers 1 to 99 go that would count as D2.