Logic Question (ladders and monkeys)

Question

In a room five ladders are leaning against the wall. At the top of each ladder there is a banana, and at the bottom of each ladder there is a monkey. Some of the steps of the ladders are connected with ropes, but to each step at most one rope is tied. Each monkey starts to climb up on its ladder in such a way that every time it finds a rope end, it climbs to the other ladder via the rope, and continues to climb upward. If two monkeys meet in the middle of a rope they can climb around each other. Whenever a monkey reaches a banana, it eats it. At least how many monkeys are going to have a banana ?

Answer 1

Every rope only swaps two monkeys, so after each rope there is still one monkey per ladder. All the monkeys will have a banana!

Answer 2

We need to make two assumptions in order to answer this:

1. No rope can be tied to a step with a banana such that a monkey can eat a banana and then cross to a different ladder.
2. No rung can have multiple ropes tied to it. With these assumptions, every single banana is accessible to one and only one monkey. To illustrate this, we can transform the ropes and ladders with a mathematically identical system. If we break every ladder at every point where a rope connects (duplicating the rungs with ropes), we can reassemble each of these ladder segments so that the rungs connected by ropes have their continuations swapped (i.e. instead of crossing between ladders, we view the monkey path as continuous).

If assumption 2 holds, every single path is well defined (there are no forks). If assumption 1 holds, our transformation keeps bananas at the top of the five main ladders. Our transformation can create new ladders (if the ropes allow a section of a ladder to be skipped completely, for instance with a rope from rung 4 to rung 7 of the same ladder with no rope connections on rungs 5 or 6), but no monkey or banana can be on these new ladders. To prove that no monkey or banana can be on these new ladders, we point out that these new ladders are from a segment with a rope attachment on top (so there can't be a new banana), and that under our transformation, monkeys do not cross between ladders.

If assumption 1 does not hold, then our transformation does not lock bananas at the tops of ladders. We can therefore construct a rope and ladder system where one monkey earns multiple bananas and at least one monkey gets no bananas.