The last ball is white.
The number of white balls in the urn is an odd number at the beginning of the process. On each step, two balls are tentatively removed from the urn. But it is never the case that this results in exactly one white ball being permanently removed; if a black and white are removed, the white is returned. If two whites are removed, both are taken.
In other words, if the number of white balls was odd before a move, it is still odd after the move. Hence, since we started with an odd number of white balls, we must end with an odd number of white balls. Since we only have one ball left, that must be white.
Thanks to Jim Fink for beating me to the solution!
This puzzle comes from
A K Dewdney,
The Tinkertoy Computer and Other Machinations,
Freeman, 1993.
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