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Originally Posted by diobsidian
You have 12 identical-looking coins. One of the coins is a fake, yet you do not know which one it is nor if it weighs more or less than the other 11. Using 3 turns with a regular balance scale, determine which coin is the fake and if it weighs more or less than the other 11.
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The details are simple but hard to explain without graphics. Label the twelve coins A through I. Weigh ABC against DEF, then ABC against GHI. The results will tell you which group of three is different and whether it is heavy or light. Then you weigh any two of the different group's coins against each other to find out which is the odd one; if they match, the third one is the different one.
if ABC < DEF and ABC < GHI, one of ABC is light
if ABC < DEF and ABC = GHI, one of DEF is heavy
if ABC = DEF and ABC < GHI, one of GHI is heavy
if ABC = DEF and ABC > GHI, one of GHI is light
if ABC > DEF and ABC = GHI, one of DEF is light
if ABC > DEF and ABC > GHI, one of ABC is heavy
The other three combinations are not possible if exactly one coin is different.
Once it's narrowed down to three coins, just compare two of them. If they don't balance, you already know whether the fake is heavy or light; just choose that one. If they do balance, the other must be the fake.
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Originally Posted by diobsidian
There is a tree with exemplary fruit in the middle of a courtyard that a farmer has decided must be guarded from people trying to take the fruit. There are 7 circular fences around the tree with a guard at each. Wanting to get at the fruit so badly, you go up to the first guard and tell him that if he lets you through, when you return you shall give him half of all the fruit you have taken, but then the guard must return one piece to you. You continue to bribe every guard you reach until you have reached the tree, whereupon you take the fruit and upon leaving fulfill your deal with all of the gaurds. How many pieces of fruit did you take?
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If I take two, then I can give one (half of my two) to each guard and according to the agreement he'll give it back to me, leaving me with the two I took.