Twelve sacks of grain and one that is short

Question

A miller has twelve sacks of grain that should all weigh the same. One sack was filled slightly short, and the miller knows it is lighter but not which one. There is a large two-pan balance that can hold several sacks on each side. What is the fewest number of weighings that guarantees the light sack is found? Describe the weighings. Be careful: this is not the famous twelve-coin puzzle, because here you already know the odd sack is lighter.

Accepted Answer

Three weighings. Lower bound. Each weighing has three outcomes, so two weighings distinguish at most nine possibilities. There are twelve suspect sacks, so two weighings cannot always work. Three weighings distinguish up to 27, so twelve should be comfortably possible. Procedure. 1. Put 4 sacks on each pan and leave 4 aside. If a pan rises, the light sack is among those 4. If the pans balance, it is among the 4 left aside. Either way you are down to 4 suspects. 2. From those 4, weigh 1 against 1. If one rises, you have found it in two weighings. If they balance, the light sack is one of the r…

Twelve sacks of grain and one that is short

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