Tuesday, July 8, 2008
Twelve Marbles
You have twelve marbles that look and feel identical. Eleven of the marbles weigh exactly one ounce. One of the marbles weighs a different amount, and you do not know it is lighter or heavier than the others. You also have a scale (the Libra kind, not the digital kind.) Using no more than THREE weighs, how can you guarantee to find the defective marble?
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ANSWER:
Split the marbles into 3 groups of 4: A (1, 2, 3, 4) B (5, 6, 7, 8) and C (9, 10, 11, 12). Weigh group A against group B. If they balance, then you know the defective marble is in group C, and you know that marbles in groups A and B are both good. Weigh 9 and 10 in group C against 1 and 2 from group A. If they match, then weigh 11 against 1. If these match, then 12 is the defective marble. If these do not match, then 11 is the defective marble. If 9 and 10 do not match, then weigh 9 against 1. If these match, then 10 is the defective marble. If these do not match, then 9 is the defective marble.
Now back to the initial weigh of A and B. If these do not match, keep track of which marbles went in which direction. For example, if side B went down and side A went up, that means that either a light marble is in group A, OR a heavy marble is in group B. For the second weigh, weigh 2 heavy marbles and 1 light marble against 2 heavy marbles and 1 light marble. If this is equal, then measure one of the remaining marbles against a good one. If it is equal, then the last marble is defective. If not, then this marble is defective. If the HHL vs HHL is unequal, and the right side goes down, then you know that either 1 of the heavy marbles on the right side is heavy or the light marble on the left side is light. Weigh the two heavies against each other. If one goes down, it is defective. If they are equal, the light one was defective
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