How to discover the relation between 4096 and 729 as result of 10 billions of simulations.

Question

First of all I'm not mathematician (really admire them), but programmer. My question come from the result or analysis from 10 billions simulations on an specific work/task.

I found that a case with 4096 possible outputs, the process only found 729 cases. This was unexpected to sense common on the task itself, so I repeat the simulation more than 10 billions times, on different ways and different RNG (random number generators) and still the numbers were the same.

My initial thought (since I'm not mathematician) was divide 4096 by 729 with no significative result (at least for me). But finally I discover that 729 is 3^6, and as 4096 is 2^12 there should be something that make a relation between this (something that I don't know and I think I wont never discover myself because I just don't know how to begin)

So that's the reason why my question is so odd, but I'm looking for an answer on how this task can resolve only 3^6 of 2^12 cases without any error margin.

Thanks in advance and sorry for my english. Please take consideration that I'm still trying learn rule of three and I'm 42yo

Answer 1

Per OP's request:

It may be significant that $4096 = 2^{12} = 4^6$, because we may then be able to interpret the situation broadly as follows:

Suppose that the simulation involves six successive selections, each of which nominally has four possible choices. There are then $4^6 = 4096$ possible combinations in all. However, if in each selection one choice is somehow accidentally disabled, then there will only be $3^6 = 729$ combinations that will actually be generated.

For instance, there are $4096$ six-digit numbers using only the digits $0, 1, 2, 3$. But if for some reason the digit $0$ is not selectable, then there are only $729$ six-digit numbers that will ever be produced.

More information about the task being simulated would make it more apparent what the situation actually is, of course.