In the puzzle Barrels Full of Pellets you were asked to find a way to discover which single barrel contained pellets that were slightly heavier than all the other barrels. This brain teaser is based on that one, but is considerably more difficult:
You are presented with ten barrels of pellets, but this time you are told that some or all *may* have 2 gram pellets and the rest have 1 gram pellets. You are given a scale for measuring and you are only allowed to take a measurement one time.
How do you find out which of the ten barrels have 1 gram pellets and which of the barrels have 2 gram pellets?
Tuesday, January 29, 2008
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Number the barrels from 1 to 10.
Take 3^(n-1) pellets from each barrel and put them on the scale (so 1 pellet from barrel 1, 3 from barrel 2, 9 from barrel 3 and so on until you take 19,683 pellets from barrel 10, whew!). Weigh the pellets (lets assume they weigh 36,925 grams).
Then start with barrel 10. Divide the total weight first by 3^(n-1) and round down to the nearest integer. So, for barrel 10, divide 36,925 by 19,683. This yields 1.876 which is 1 rounded down. This means barrel 10 has 1-gram pellets. Now, multiply the size of the pellets (1 gram for barrel 10) times the number of pellets removed from barrel 10 (19,683) and subtract from the total weight (36,925 - 1 * 19,683 = 17,242). Move to barrel 9. Divide 17,242 by 6,561 which is 3^(9-1). The result is 2.628 which rounded down is 2 which means barrel 9 has 2-gram pellets. So, multiply 2 * 6,561 and subtract from 17,242. This step yields 4,120. For barrel 8, divide 4,120 by 2,187 which is 3^(8-1). The result is 1.884 which rounded down is 1 so barrel 8 has 1-gram pellets. Continue this process and you should find that in this case, barrels 2, 4, 5, 7 and 9 have 2-gram pellets, the rest have 1-gram pellets.
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