The Last Ball - Futility Closet

An urn contains 75 white balls and 150 black ones. A pile of black ones is also available.

The following two-step operation occurs repeatedly. First we withdraw two balls at random from the urn, then:

If both are black, we put one of them back in the urn and throw the other away.
If one is black and the other white, we put the white one back and throw the black one away.
If both are white, we throw both away and put a black ball from the pile into the urn.

Because the urn loses a ball at each step, eventually it will contain a single ball. What color is that ball?

	Select Click for Answer
The number of white balls in the urn remains unchanged in the first two cases, and it drops by two in the third. That means its parity never changes — since it was odd to begin with (75), it must remain odd throughout, and the last ball can’t avoid being white. By the same principle, if the urn had originally contained an even number of white balls, the last ball would be black. From the 1983 Australian Mathematical Olympiad, via Ross Honsberger, From Erdös to Kiev, 1996.