The Scrambled Boxtops Puzzle

Boing Boing reprinted this puzzle from the book My Best Mathematical and Logic Puzzles by Martin Gardner.

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, BW, WW – but someone 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?

It’s not difficult to figure out if you can visualize the boxes in front of you (or just look at the picture). It wouldn’t be hard to make this a real world puzzle, either. Give us your answer in the comments!


Newest 5
Newest 5 Comments

You need only ONE draw.
You must draw from the box now labeled (actually, now MIS-labeled) as "BW." WHATEVER color marble you draw makes it either the WW or BB box. But actually all THREE boxes have been determined. Since the other two boxes are also mislabeled, the one with the XX-label (either BB or WW) that matches your color MUST be the actual-BW box (since it CAN'T be the actual-BB box if your draw was a WHITE, and it CAN'T be the actual WW-box if your draw was a BLACK).
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"It gets a bit harder if the labels are just randomized." is just a neat euphemism for, "impossible" on the basis of the given information (as you don't know the exact number of marbles in each box)... (e.g. you have to take at least 4 draws in case the boxes each only hold two marbles)
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1 Draw.
The key is each box must be labeled *incorrectly*. It gets a bit harder if the labels are just randomized.
First, you draw from the BW Box. Since it can't be BW, it has to be either BB or WW, so whatever color you get, put that label on the box.
The box you just identified is correctly labeled, so set it aside. So you now have 1 Colored box, one Blank box, and the BW label. Well, the Colored box doesn't match its label, so move that to the blank box, set it aside, and put the BW label on the remaining box.
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