I was asked a question about the game "Satisfactory" and how to split an output into $5$ equal parts using "splitters" with one input and 2-3 outputs and mergers that accept 2-3 inputs.
A splitter accepts input from one conveyor belt and its output feeds two or three other conveyor belts. A merger accepts input from two or three conveyor belts and its output feeds a single conveyor belt.
$$\frac{a}{2}+\frac{b}{3}=\frac{c}{5} \implies c = \frac{5 a}{2} + \frac{5 b}{3}$$
I'm out of ideas. I'm beginning to think it's not possible. How do I split something into $5$ equal parts when I can only split/merge things by $2$s or $3$s?
There's a simple way to solve this problem. As noted in the comments, there are no integer solutions to
$$2^x3^y=5^z$$
due to the fundamental theorem of arithmetic, so it's impossible to represent $\frac15$ using finite sums of multiples of $\frac12$ and $\frac13$.
However, we can "cheat" by using the sum of the geometric progression: $$\frac15 = \frac16 + \frac1{36} + \frac1{216} + \frac1{1296} + \dots$$
We can implement this with the mergers and splitters by using one 3-splitter and three 2-splitters to do a 6-way split, then feed one of those outputs back into the input stream. Here's a diagram: