Alice and Bob Marble Fight - Part 3

Before any collisions there are 30 marbles travelling towards Bob.

A collision reverses the direction of two marbles, and so the number of marbles travelling towards Bob does not change.

Bob therefore receives 30 marbles - he gets all his own back plus $\boxed{10}$ red marbles from Alice.

Assume Alice is on the left end and Bob is on the right end.

Do this problem first to get that Alice receives $20$ marbles and Bob receives $30$ marbles. However, note that the relative order of the marbles never change; if a marble X is to the left of a marble Y before collision, then after the collision marble X remains to the left of marble Y. Since at the beginning the leftmost $30$ marbles are red (Alice's), at the end, the leftmost $30$ marbles will still be red. But the leftmost $30$ marbles are Alice's $20$ and Bob's $10$ , so Bob gets $\boxed{10}$ red marbles. (The remaining $20$ red marbles remain with Alice.)

if we assume all blue marbles collide with red marbles. there shall be 30-20=10 marbles which reach Bob.

Unless I'm missing something obvious, shouldn't the solution here be "any number between 10 and 30"? After all, the question doesn't guarantee that there will be any collisions at all. Or perhaps the question should read "what's the least number of red marbles Bob ends up with?"

Let the scenario when all the Red marbles are moving towards Blue and vice-versa as neutral scenario. Let's number the marbles chronologically in the order in which they are thrown. e.g. the first Red marble thrown by Alice is numbered first Red marble.

When the first Red marble collides with first Blue marble, they both bounce back and have collision with second marble from their own pack and return to the original direction.

Now the second Red and Blue marbles are moving in opposite direction (compared to the original direction).

The second Red (Blue) marble collides with third Red (Blue) marble. Now the second Red (Blue) marble returns to the original direction and the third Red (Blue) marble is moving in opposite direction.

This process continues and the last Red and last Blue marble each return to Alice and Bob respectively. At this moment, we reach the neutral scenario.

This entire process repeats till all (20) the Blue marbles return to Bob. An equal number (20) of Red marbles returns to Alice. The remaining 10 Red marbles reach Bob.