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My Introspective

by Laurus Nobilis
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Paradox of Collatz Conjecture – Mathematical Problem or a Spin?

 

Collatz Conjecture is typical mathematical approach of creating paradox by linking two unknowns into the infinite loop. This problem is a total spin or total misinterpretation.

 

Posted: Oct 2016


People like puzzles. Some people like solving puzzles because that represent some type of brain gymnastics. Also, solving the puzzle represent the recognition of mental abilities and patience. Both, exercise and recognition helps self-esteem of every individual. Solving of problems does have a clear motive.

Making of puzzles also requires a considerable mental exercise, but also much higher and broader knowledge, imagination and experience. But what about other motives? Puzzle makers also wants public recognition of their work. This recognition may be moral, material or other recognition that can bring different gains to puzzle maker in the future.

But what happens when the puzzle cannot be solved? Or the puzzle is made for future mathematicians, because the mathematicians of present are not ready yet. This can be frustrating, since wasting of time without result is nothing than anybody likes.

The worst example of puzzle is the one where the design of the puzzle does not allow solution to be found, because there is no any. One of such mathematical puzzles is the Collatz's Conjecture.

Collatz Conjecture

Take any positive integer number and apply one of two given operations:

EVEN NUMBER – Divide by 2
ODD NUMBER – Multiply by 3, then add 1

Process the resulting number the same way multiple time
Final result is always number 1

The Collatz's Conjecture says that for any given positive integer we should apply mathematical operation. If given number is even, we need to divide it by 2. If it's odd, we need to multiply it by 3 and then add 1. Now we need to repeat the process with our new number. If you keep going, you'll eventually end up at 1. Every time. This is true.

Mathematicians have tried millions of numbers and they've never found a single one that didn't end up at 1 at the end of the process. Also, they like to claim that they have never been able to prove that there isn't a special number out there that doesn’ lead to 1. They suspect that it is possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1.

OK, all this is true. But the question is: What is the mystery about this problem, when the problem itself gives the explanation? The purpose of the model is to get number 1 at the end, not some other number.


The problem itself shows that we are adding 1 after odd number is tripled. If you add 1 to odd number, you get an even number. If that even number is the power of 2, then you will get 1 at the end.

If you have even number that is not power of 2, then you will get odd number, which you will have to triple and increase by one, but again you will get even number. Eventually, you will hit the power of 2 and the final result is 1.

The same way the odd number will become even and will eventually reach 1.
So the problem itself is defining the solution. The purpose of +1 in this process is evening of odd numbers and fishing of number that is power of 2.

Mathematicians really loves mysteries. Even if they cannot find the number that never reaches 1, they want to fuel the fire by stating that despite they didn't find desired number, they cannot prove that such number does not exists. So, they want double mystery. They cannot prove something, but also they cannot prove opposite. This is typical mathematical approach of creating paradox by linking two unknowns into the infinite loop.

So, Collatz Conjecture is very interesting puzzle at first glance, but it is obviously a spin, or people are misinterpreting the meaning of Collatz algorithm. It is amazing that so many people, including mathematicians, hooks on every math problem, writing a books, posting YouTube videos, creating mystery about something that is (intentionally or non-intentionally) artificially self-generated problem.

Simply, the algorithm is designed to reach 1, regardless of positive integer used. Algorithm does what is designed for – to reach 1 every time! Algorithm is not designed to find exceptions. So why are people looking for something which is not the scope of this algorithm?  

Con-Collatz Rules:

  1. The problem is artificially self-generated because of "+1" during the process

  2. Even numbers that are power of 2 are always heading to 1 (... 16, 8, 4, 2, 1)

  3. Even numbers that are not power of 2 are becoming uneven, then even ... until they hit the number that is power of 2

  4. Uneven number becomes even, thou reaching rule 2. through one or more iterations

  5. Theory that some magical positive integer number that never lead to 1 exists outhere, waiting to be discovered, can be possible only if some fourth group of number exists (apart of numbers from rule 2, 3 and 4.)

Power of 2

In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with number two as the base and integer n as the exponent.

21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1024

 

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