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, ...

The list is redundant once you perceive that the transformation of the 3-digit number 100a+10b+c into a^2+b^2+c^2 produces always a lesser number, so in few steps a 2-digit number must be reached, - from this moment on the reduction does not necessarily hold and the process should be examined.

Proof: Comparing 100a+10b+c and a^2+b^2+c^2 we get:

100a-a^2>99 ....FROM 99 TO 819

10b-b^2>=0 ''''0 TO 25

c-c^2>=-72'''''-72 TO 0

so the number goes down by at least 27(109==>82 ) and at most by 844 (950==>106 and 951==>107 & later below 100..).