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## Color of the last ball

In a large urn there are \(w\) white balls and \(b\) black balls. Beside the urn is a big pile (infinite number) of black balls. Now, we do the following. First, two balls are drawn at random from the urn and

- if they are both black, one of them is put back and the other is thrown away,
- if one is black and the other white, the white one is put back and the black one is thrown away,
- if they are both white, they are both thrown away and a black ball from the pile is put into the urn.

Therefore, whatever the case, at each stage two balls removed from the urn and one is put back, thus reducing the number of balls in the urn by one. Eventually, then, the urn will reach the point of containing just a single ball.

What is the color of this last ball ? Does it depend on \(w\) and \(b\) ? If so, what is the dependence ?

**Source:**folklore