Alice and Bob Marble Fight - Part 1

Let Alice=A and Bob=B . So when A rolled 30 marbles from the left side and B rolled 20 from the right side, The 30th marble of A collided with 20th marble of B. So, the 30th marble turned left and collided 29th marble of A and again turned right. Similarly 29th marble turned left and collided 28th marble and again turned right. This went on up to when 2nd marble turned left, collided the first one and again turned right. Same happened with B but in opposite direction. At the end of 1 set of collision 1 marble remained with A and 1 with B. Now, 29 marbles are going from A to B and 19 marbles are going from B to A. This continues till only 1 marble goes from B to A and 11 goes from A to B. Once the last marble of B collides with 11th marble of A it returns B and the process repeats in A, where the 1st marble goes to A and the remaining 9 marbles of A go to B. Hence, at the end, A has 20 marbles and B has 30

In the beginning, we have 30 marbles going towards Bob and 20 marbles going towards Alice.

Before a collision, we see that one marble has to be moving Alice and one marble has to be Bob, since marbles won't collide if they are going in the same direction. After the collision, the same situation exist, i.e. one marble is moving towards Alice and one marble is moving towards Bob, since the marbles bounce back.

So the number of marbles going towards each of them remains invariant under the collisions.

Since 20 marbles were going towards Alice in the beginning, 20 marbles must go to Alice in the end irrespective of the number of collisions in between.

After a series of collisions, 1 marble from both the ends will go back to their original owners. Suppose we have 30 coming from the left and 20 from the right.

The number of marbles start decreasing together, and, when the right side has 0 marbles,i.e., Bob has his 20 marbles back, there will be 10 marbles on the left , i.e., Alice will also have 20 of her marbles back.

So now we have 10 marbles on the right and no marbles on the left and hence these 10 marbles will keep moving towards the right in sync and will end up with Bob

Thus Alice will have $20$ marbles left.

since bob has only 20 marbles only the 20th marbel thrown by alice will come back and the rest will go to bob hence alice will have only 20 marbels

The no. of marbles will interchange between the two ... ! Because the marbles of higher owner( in terms of quantity) will collide more often to change directions .. !!