Back in the Days in School I remember a teacher told us about this Problem.
Consider the following operation on an arbitrary positive integer:
- If the number is even, divide it by two.
- If the number is odd, triple it and add one.
As result you will end up at 1, which produces 1, 4, 2, 1 as a period.
There are hundreds of People out there in Universities and High Schools who try to proof that there
is no point to start from that breaks the rule.
How hard can it be to proof? There can't be any number!
To be honest - People try to proof the whole algorithm. You just have to proof, that an odd number time 3 plus 1 cannot create
a number which result is odd after taking it 3 times plus 1.
((((((n % 2 != 0) * 3)+1) % 2 != 0) * 3) +1) %2 == 0
To finish it from my point of view:
Is n even there is a smaller change to end on an odd number than ending on an even.
That results in the maximal possible sum of (((n *3 )+1) * 3) +1) that will be even in any case.
This will divided by two and has a new change by 50% to be odd.
Since there is a change of 25% to hit a odd number at it has to peak up first but must end in 1 after several tries because the rules
for even numbers is always 4:1 because of 50% of every number is even and getting an even number after the odd-rules is higher then getting an odd one.
In fact - it is just as rolling the dices long enough.
The peaks are a manoeuver, which distract from the smaller opportunity of growth. That is the thing that tricks the people. It isn't a wave. It is an artificial Peak but at last, the changes are always 1:4
that you get an even number.
Just for fun - you can write:
n = {
if n ≡ 0 (mod 2) => n = 2/n
OR
if n ≡ 1 (mod 2) => n = (3n+1)/2 ≡ 0 (mod 2)
}‾ = 1
This should proof it :)
But that wasn't why I post this.
What makes me writing is that it looks like Forex or Gambling. The Function feeds you with a few points of success that keeps you playing.
The only thing that makes you playing more is how much you feed the Slotmachine. Am I the only one that see this?
If I look closer - The Collatz conjecture is the best formalism that explains economics.
[1]
Wikipedia: Collatz conjecture