Htydjtk

Our team has created 9 math puzzles with ranging difficulty. Let me share one of them.

At the beginning a number is given to you. Your objective is to get a different number by making choices. Here's the pseudo code.

Let's say your number is $3$ and you need to reach $56$.

Start

If your number is ODD.

1. Mr. Black multiplies it by $3$




2. Then you should add $2$ or add $3$




3. Go back to Start with your new number

Else (your number is EVEN)

1. Mrs. White offers you to divide it by $2$



2. 
   

   If you accept the offer

   
   
   
   
   

   1. Mrs. White divides your number by $2$

   
   

   2. Go back to Start with your new number

   
   
   
   

   Else

   
   
   
   
   

   1. Mr. Black multiplies your number by $3$

   
   

   2. Then you should add $2$ or add $3$

   
   

   3. Go back to Start with your new number

   
   
   
   

If you've managed to get $56$ in Start step of this algorithm, then puzzle is solved.

As an example, here's the evaluation

$3 to 9 to text{by adding 2} to 11 to 33 to text{by adding 3} to 36 to text{by accepting offer} to 18 to text{by declining offer} to 54 to text{by adding 2} to 56$

Here are some puzzles for you to think
$$3 to 1$$
$$65 to 9$$
$$15 to 128$$

My hypothesis is: You can reach from any natural number to any other natural number (so there can be infinite pack of "interesting" puzzles).
What I have noticed, the more these numbers are far from each other, the harder it is to reach to solution. The puzzle is inspired from well known problem :)

I have 3 questions:

1. Can you solve all of them? :)




2. Is there a strategy or an algorithm, by which you can always reach to solution with any given source and target numbers?




3. Can you prove or disprove my hypothesis?

You can try this puzzle with a more interactive way here (since level 6)

P.S: This is my first puzzle question. Hopefully the formatting is correct.

[EDITED to add: When I wrote this, the original statement of the problem had an error in it. That's now been fixed but what follows hasn't, so far, been updated accordingly.]

The most interesting of the questions originally asked is the third: is it possible to get from any number to any other number by this process?

The answer is

no, there are some $x,y$ for which you can't do $xrightarrow y$.

Here's a way to see that.

Call a number good if it's a multiple of 3 OR it's a power of 2 times a number of the form $6k-1$. Suppose $n$ is good; then if it's odd you can replace it with $3n+2$ which you can easily see is of the form $6k-1$, or with $3k+3$ which is a multiple of 3. And if it's (good and) even you can replace it with $n/2$ or $3n$, and clearly both of these are good since $n$ is. Therefore, if you start with a good number then you can only ever reach good numbers. Since some numbers are good and some aren't, it follows that we can't reach every number from every number.

I wonder, though, whether either I have misunderstood the question or OP hasn't asked quite the question they intended to, because

in particular this shows that you can't get from 3 to 1, which is one of the specific puzzles posed; I wouldn't expect OP to be conjecturing that you can get from any number to any other if they'd tried to get from 3 to 1 and failed...