Question

We have a bag filled with 201 marbles, of which 100 of them are blue and 101 of them are red. Every turn, we remove 2 marbles from the bag. If the two marbles are of the same color, we remove the two marbles but add a blue marble into the bag. If the two marbles are of different colors, we remove the two marbles and add a red marble into the bag. What is the color of the last marble in the bag? [Need step by step detailed answer].

Answer 1

At each step, there are 3 different cases:

1) We get two blue marbles: In this case, number of blue marbles reduce by 1, number of red marbles stays the same.
2) We get two red marbles: In this case number of blue marbles increase by 1, number of red marbles reduce by 2.
3) We get one red and one blue marble: In this case the number of blue marbles reduces by one, number of red marbles stays the same.

In all three cases, we can see that the number of red marbles is reducing by 2 or remains the same. Since we 101 red marbles, there is no possible way to get rid of the red marbles in the steps of two. So at least one red marble will always remain. This gives us that if there is only one marble remaining, it has to be red. Hence the red marble is the last marble.