5. Scrambled Box Tops

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IMAGINE THAT YOU have three boxes, one containing two black marbles, one
containing two white marbles, and the third, one black marble and one white
marble. The boxes were labeled for their contents—BB, WW and BW—but
someone has switched the labels so that every box is now incorrectly labeled.
You are allowed to take one marble at a time out of any box, without looking
inside, and by this process of sampling you are to determine the contents of all
three boxes. What is the smallest number of drawings needed to do this?

The answer

You can learn the contents of all three boxes by drawing just one marble. The

key to the solution is your knowledge that the labels on all three of the boxes are
incorrect. You must draw a marble from the box labeled “black-white.” Assume
that the marble drawn is black. You know then that the other marble in this box
must be black also, otherwise the label would be correct. Since you have now
identified the box containing two black marbles, you can at once tell the contents
of the box marked “white-white”: you know it cannot contain two white
marbles, because its label has to be wrong; it cannot contain two black marbles,
for you have identified that box; therefore it must contain one black and one
white marble. The third box, of course, must then be the one holding two white
marbles. You can solve the puzzle by the same reasoning if the marble you draw
from the “black-white” box happens to be white instead of black.

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