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4,4,4,4.... (Posted on 2007-07-31) Difficulty: 3 of 5
I chose a certain sequencing rule, and I told my friend no matter what number he named, it would eventually end up at repeating 4s.

At first he chose 5, which ended in 4, 4, 4... Then he tried 18, which met the same fate. Even numbers like -42 and Pi ended in 4, 4, 4...

Still trying to outdo me, he started to get really creative. He tried 63424563562324 ... same. He even tried a googolplex and Avogadro's number. But alas all the numbers he tried ended up in 4,4,4. Finally he conceded.

What's the sequencing rule I use? Should he have conceded? Can you find any starting terms which don't end in 4, 4, 4...?

See The Solution Submitted by jduval    
Rating: 4.3333 (3 votes)

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Solution Might be the solution | Comment 1 of 8
Consider the following sequence,
n,abs([n]){=m},d([m/2]){=x},d([x/2]){=y},d([y/2]),....
where d(n): No. of divisors of x(including negative integers too)
[n]:greatest integer less than or equal to n
abs(n):n*1 if n>0, n*(-1) if n<0 and 0 if n=0.
There will definitely be a case where [k/2] is prime. From then
d([k/2])=4,d(4/2)=4{-1,-2,1,2}, and this repeats.
The case where it doesn't end in 4's is when starting term in the range
[-1,2), because d(0) is not known and d(1)=2{-1,1} which repeats itself.
Now consider this
n,[abs(n)]{=m},d([m/2]){=x},d([x/2]){=y},d([y/2]),....
Values for n for which it will not satisfy (-2,2)

Edited on July 31, 2007, 10:01 am

Edited on July 31, 2007, 11:18 am

Edited on July 31, 2007, 11:19 am
  Posted by Praneeth on 2007-07-31 09:55:39

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