Trying to find or similar problems (objects travelling across slots)

[aRiver]

I'm trying to find a way to approach a problem:

No two objects can occupy the same slot.
Objects can travel 1, 2, or 3 slots per period.
There are 18 slots.

That is really a very trivial case of a more complex problem. My question is does this sound like similar to any existing problems in mathematics? I'm trying to find a way to analyze the problem (matrix ? modulo arithmetic ?)

a simple case follows

three objects released at the same time, with rate 1, 2, 3.
step1 : 1,2,3 occupied
step2 : 2,4,6 occupied
step3 : 3,6,9
step4 : 4,8,12
step5 : 5,10,15
step6 : 6,12,18
step7 : 7,14
...
...
...

Eventually I want to analyze a way to, say, maximize objects of any type which traverse the slots over a given number of steps, or how changing number of slots effect the problem. Rather than ask for a solution or analysis, I was wondering if anyone could point me to a similar problem, because I feel like this is a more complex version of a solved problem.

 

[Stephen Tashi]

Eventually I want to analyze a way to, say, maximize objects of any type which traverse the slots over a given number of steps, or how changing number of slots effect the problem.

You haven't defined a mathematical objective yet. It isn't clear what "traverse the slots" means. It isn't clear whether the objects are each to be treated as distinct from each other.

One way of looking at this is that there are a set of possible "states", each state being a set of 3 numbers that tell which cells are occupied ( e.g. (1,2,3), (2,4,6),(1,7,9) .. etc.) You can define a matrix A by setting $A_{i,j} = 1$ iff it is possible for the system to change from state $i$ to state $j$ and $A_{i,j} = 0$ otherwise. The possible changes the system can make in $n$ steps is given by $A^n$. The matrix $A$ specifies the edges in a graph..

You'll probably get better advice if you explain the actual problem that you are dealing with. Asking how to apply mathematics to a problem is an interesting challenge. On the one hand, if you write a long post filled with details, you risk making it so tedious that nobody will respond. On the other hand, if you attempt to extract the essential details and summarize them, you are exercising your own judgment about which facts are the mathematical essentials, so you are removing that function from any of your advisors.