Scrambled Box Tops

Imagine that you have three boxes , one containing two black marbles , one containg 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 ?

You can learn the content 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 marked “black-white”. Assume that the marble drawn is black. You know then the 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 drawn from the “black-white” box happens to be white instead of black.