By Erika Schwibs
The Collatz Conjecture is like a game, Odd numbers against Even, and Evens have the built-in advantage. At most, the Odd “team” multiplies a number by 3. That seems to accomplish a lot as the conjecture produces extremely large numbers. But the Even team can divide by the largest power of 2 that will go in to a number, and all of those divisors, except for 2, are larger than 3. Or another way of putting it is that the Odd team can only go once per “turn,” when it merely multiplies by 3, while there is no fixed limit for how many times the Even team can divide by 2 per “turn” — and that means that it often essentially divides by extremely large numbers.
And there’s still another way of looking at the Collatz “game” situation: Evens “win” 100% of the rounds. When 3x+1 produces an even number, Evens get to divide again (essentially, dividing by 4). But if an odd number is produced, then that odd number is “corrected” to an even number. That number will be about 3x larger than the prior one, taking Evens somewhat further away from their goal of getting to 1, but 100% of the time, the corrected number, despite being about 3 times bigger, will be even. So, in essence, the prior number wasn’t basically tripled, but only multiplied by 3/2 [with 0.5 then added on, if memory serves me). And 50% of the time, the number produced will be divisible by at least 4; 25% of the time, it will divisible by at least 8; etc. In the face of numbers that can be divided by any power of 2, without any limits, being limited to multiplying by 3 isn’t much of a countermove.
Perhaps our human perspective and limitations somewhat hinder our thinking about the Collatz Conjecture, as the numbers involved get too big for even computers to reach. But no matter how large the number, they still are produced just one number at a time, and still follow the same 1, 2, 3…and 49, 50, 51…Every other number will be even and divisible by 2, every fourth will be divisible by 4, every sixth by 2 and by 3, every 8th by 2^3 and every 16th by 2^4. We can be sure that the “orderly order” of numbers doesn’t become freaky somewhere, no matter how large the number, even those presently beyond human reach. And 3x+1, producing only even numbers, can’t produce any prime numbers. And no matter how big the numbers get when produced by 3x+1, they always contain many combinations of powers of 2 as factors. One extremely large even number may only divide by 2 once, but the combined orderliness of the number system and the Collatz Conjecture “rules” ensure that a “player” will often be dividing by power of 2 greater than the number 3, and sometimes very large powers of 2. That’s inherent in even numbers themselves, and those matter more in the Collatz conjecture simply multiplying by 3.
It’s possible, though, to even the playing field of the Collatz game. Suppose all the rules of the Collatz Conjecture game are kept the same, but one: rather than giving Evens an unlimited number of moves, it’s limited to one move at a time, just like the Odd team is.
For instance:
3(1)+1=4. 4/2=2. 2*3+1=7.
Under the game rules, you can’t divide an odd number by 2, but you also can only multiply by 3 and add 1 one time in a row, so the “game” ends here. For every number, the game is uniquely different, just as the standard Collatz Conjecture “game” is.
3(2)+1=7
7*3+1=22
22/2=11
11*3+1=34
34/2=17
17*3+1=52
52/2=26
26*3+1=79
Game over at that point as, again, you can’t “3x+1” again, or divide by 2.
Perhaps there are some relationships between the Collatz Conjecture and this variation on it.
Faith Presses On 