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Two happy ends (Posted on 2011-03-07) Difficulty: 3 of 5
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, ...

Ex1: 12345,55,50,25,29,85,89,145….. etc
Ex2: 66,72,53,34,25,29,85,89,145…
Ex3: 91,10,1,1,1…..

  Submitted by Ady TZIDON    
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Solution: (Hide)
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 .

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre(7): NOT enough by GamerAdy TZIDON2011-03-09 19:46:55
Hints/Tipsre: How the numbers get there- I EXPLAINAdy TZIDON2011-03-09 19:43:15
How the numbers get thereCharlie2011-03-09 15:39:48
re(6): NOT enough by GamerCharlie2011-03-09 15:20:18
re(5): NOT enough by GamerAdy TZIDON2011-03-09 07:17:33
re(4): solution NOT enoughGamer2011-03-09 04:46:40
re(3): solution NOT enoughJer2011-03-08 10:08:47
re(2): solution NOT enoughGamer2011-03-08 05:17:54
Hints/Tipsre: solution NOT enoughAdy TZIDON2011-03-07 17:57:23
SolutionsolutionCharlie2011-03-07 15:15:40
Hints/Tipssolution enoughJer2011-03-07 14:32:20
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