Picking balls from bags (solution)

This is the solution to this puzzle: Picking balls from bags

You only need to pick one ball from the bag labelled black+white.

Suppose you picked a black ball from the bag. (The deduction will be the same even if you picked a white ball because of symmetry.) Because the label is wrong, it is certain that the other ball in the bag is a black ball. Thus, out of six balls, 3 white and 3 black, we have identified the location of 2 black balls. The third black ball must be either in the bag labelled white+white or black+black.

However, because the labels are all wrong, the bag labelled white+white must contain a black ball, i.e. the remaining black ball of the three. Therefore the contents of this bag has also been identified: a black ball and a white ball.

The last bag, labelled black+black, naturally, contains two white balls.