A Conjecture Equivalent to the Collatz Conjecture

Consider an organization where we have a linear hierarchy among the employees. Each employee can be in one of 3 different “state of mind”: sleepy, confused, or motivated. Furthermore, each employee can get a gentle-nudge or big-push (that is generated by their supervisor), and each employee can generate either a gentle-nudge or big-push for their subordinate.

Formally, an employee is a Mealy machine [3, 2] with 3 states (sleepy, confused, motivated) and two input (and output) symbols (gentle-nudge, big-push). An organization is formally an ordered list of employees. Consequently, the state of an organization is a list of states of its employees; for example, a possible state of an organization with 3 employees is $[{\tt{motivated}},{\tt{confused}},{\tt{sleepy}}]$.

Every morning the top member of the organization gets a gentle-nudge, which is propagated down through the day to all the employees in the hierarchy based on the following rules:

1. 1.

   When a sleepy gets a gentle-nudge, they continue to remain a sleepy and generate a gentle-nudge for the next person.

2. 2.

   When a sleepy gets a big-push, they turn confused and generate a big-push for the next person.

3. 3.

   When a confused gets a gentle-nudge, they turn sleepy and generate a big-push for the next person.

4. 4.

   When a confused gets a big-push, they turn motivated and generate a gentle-nudge for the next person.

5. 5.

   When a motivated gets a gentle-nudge, they turn confused and generate a gentle-nudge for the next person.

6. 6.

   When a motivated gets a big-push, they continue to remain motivated and generate a big-push for the next person.

If the last person in the hierarchy generates a big-push, then a new junior motivated person is hired (and added to the organization).

If at the end of the day all members of an organization turn sleepy but for possibly the last member, then the organization goes bankrupt.

We can easily formalize the six rules enumerated above as defining the transition relation and output function of a Mealy machine. We write $s\rightarrow t$ to denote that state $s$ of an organization (at the start of a day) changes to state $t$ (at the end of that day). A state $s$ is bankrupt if only its last element is possibly not sleepy.

We first recall the Collatz conjecture [1].

We now show that Conjecture 1 is equivalent to the Collatz conjecture. We use a novel representation of numbers for this purpose. Each digit in our representation is either $0,2,$ or $4$, and the place value is interpreted as in Base-3 representation. For example, the number $420$ in our new representation denotes the decimal number $4*3^{2}+2*3^{1}+0*3^{0}$, which is $42$ in base-10. We can only represent even numbers in this new representation. In this new representation, if $0$ is written as sleepy, $2$ as confused, and $4$ as motivated, then each number $n$ can be seen as a state of an organization. For example, $42$ is represented by the organization $[{\tt{motivated}},{\tt{confused}},{\tt{sleepy}}]$.

We modify the iterations in the statement of Collatz conjecture to go from even number to an even number as follows: Define $g(n)$ to be $n/2$ if $n/2$ is even, and $3(n/2)+1$ if $n/2$ is odd. Now, it is an easy exercise to see that if $m=g(n)$, and $s$ is a state representing $n$, then $s\rightarrow t$ where $t$ represents $m$. This shows the equivalence to Collatz conjecture.

There is some recent work [4] on trying to prove the Collatz conjecture using automated termination proving techniques for rewriting systems. For this purpose, one needs a rewriting system that mimics the iterations in Collatz conjecture. There is one such rewriting system in [4]. The formulation presented here can also be turned into a (different) rewrite system.