The problem:
Imagine you had a number line from 0 to infinity and plots will be made on that line based on unique plotting intervals (ex. every $16.5x$). What are the maximum amount of plotting intervals that can be used in conjunction without breaking the following rules. What are those intervals? What if the number line was only between 0 to 20?
-Plots may never repeat across intervals (ex. if one interval is $0.2x$ then $0.6x$ is not a valid interval because they repeat at $x=0.6$)
-Intervals themselves can not repeat or be staggered by adding a delay.
-Intervals can have, at most, 1 decimal place (ex. $1.5x$ and $2.0x$)
-Interval must be between 0.0 to 100.0
My question:
Is there a way to find these intervals without brute forcing all intervals, $n$, between $0.0x \le nx \le 100.0x$ that have unique multiples?
How I got here:
Hello there! Here's some explanation about how I got here. Read if you want to, it might help clarify my question if it is unclear haha .
I hope to be an engineer one day but am pretty bad at math. I was playing a video game called Factorio the other day. It got me thinking about assembly lines working in parallel and contributing the same amount of components to a collective processing area.
Then I thought about a scenario where someone only has a single conveyer line but infinite many feeders of unique feeding rates contributing components to that line and how someone would go about figuring out the necessary timing to prevent any of the feeders stopping because a component from another feeder is passing through at the same time as it was going to contribute its own component.
I considered how, if every feeder needed to feed at a different rate, the fastest feeder put at the first position would contribute the most efficiency because it can contribute most frequently and that every added feeder contributes less.
Because of that thought I came up with this question but I can't figure out what topic/field of math this type of problem would be from or how I could go about efficiently solving it.
Is there a way to do this without simply brute forcing all intervals, $n$, between $0.0x \le nx \le 100.0x$ that have unique multiples to infinity or to 20?
What about intervals where, instead of finding the maximum amount of unique intervals, you wanted to get a fixed number of unique intervals, such as 10 intervals, that plot at rates with the smallest standard deviation (most alike in efficiency contribution to the conveyers)?
I realize irl you'll just use multiples of a base number based on the # of each component needed and staggering the feeding but I was just curious about this scenario. Thanks in advance for the help! Looking forward to heading down another mathematical rabbit hole.
