Quote: Originally posted by ayenowitall on May 22, 2004
I really appreciate the attention that this mathematical sequences topic has received. Much thanks to all who have made meaningful contributions related to mathematical sequences. I hope you will all continue to offer your input on mathematical sequences. An occasional tangential remark is understandable, but this topic is really about mathematical sequences. We have a great group of people here at Lottery Post, and I'm sure that there must be considerable knowledge about mathematical sequences. I'm looking forward to learning more about mathematical sequences from those who have such knowledge.
Also, we all make typographical errors from time to time, but gross misspellings that occur repeatedly reflect poorly on the writer's intelligence and credibility. They are also quite an annoying distraction. There is a Spell Check function available.
sorry bout the O.T. posts...
here's something interesting...
Mathematical mysteries: Hailstone sequences
This problem is easy to describe but it is one of mathematics' unsolved problems.
Starting with any positive integer n, form a sequence in the following way:
If n is even, divide it by 2 to give n' = n/2.
If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.
Then take n' as the new starting number and repeat the process. For example:
n = 5 gives the sequence
5, 16, 8, 4, 2, 1, 4, 2, 1,...
n = 11 gives the sequence
11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...
These are sometimes called "Hailstone sequences" because they go up and down just like a hailstone in a cloud before crashing to Earth - the endless cycle 4, 2, 1, 4, 2, 1. It seems from experiment that such a sequence will always dventually end in this repeating cycle 4, 2, 1, 4, 2, 1,... and so on, but some values for N generate many values before the repeating cycle begins. For example, try starting with n = 27. See if you can find starting values that generate even longer sequences.
An unsolved problem is, can it be proved that every starting value will generate a sequence that dventually settles to 4, 2, 1, 4, 2, 1,...? Could there be a sequence that never settles down to a repeating cycle at all?