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, ...

1.It can be easily proven that any number over 99 produces a lesser number, the reduction being at least 25 and at most 845.
2. The above causes the chain to finally rach a single number 1 or the endless loop of the numbers mentioned in the text.
3, To show that there exist no other loops one has to examine all the numbers from 1 to 99 inclusively (Charlie tested the 999 first numbers, always stopping either at "1" or at "4".).
4, To perform a "manual" test of the 99 numbers one should not test all of them, but erase from the candidate list all the numbers of the presently inspected initial number: e.g. 99=>162=>41=>17=>50=>25=>29=>85=>89 …etc stop,- 41,17,50,25,29,85,89, … etc need not be tested .

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