There is a bag with 20 blue balls and 13 red balls. Randomly remove 2 balls from the bag:
- If they are the same color, replace them with a blue ball
- If they are different colors, replace them with a red ball
Repeat this process until there is just 1 ball remaining. What is the color of the last ball?
Each time 2 balls are removed, there are 3 possible scenarios:
- 2 blue – replace with 1 blue, so end result is 1 fewer blue ball
- 1 red, 1 blue – replace with 1 red, so end result is 1 fewer blue ball
- 2 red – replace with 1 blue, so end result is 1 more blue ball and 2 fewer red balls
Notice that across the possible scenarios, red balls can only be reduced in pairs. Since we started with an odd number of red balls, that means there must always be a red ball remaining in the bag, so it must be the color of the last ball.
(We also see that if we instead started with an even number of red balls, the last ball must be blue, because there could never be just 1 red ball remaining.)