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You have 12 coins, one of which is fake. The fake coin is indistinguishable from the rest except that it is either heavier or lighter, but you don't know which. Can you determine which is the fake coin and whether it is lighter or heavier using a balance scale and only 3 weighings?

?

 [Ref: ZLKY]


One solution is to label the coins with the letters FAKE MIND CLOT and weigh the coins in the following three combinations:

MA DO -- LIKE
ME TO -- FIND
FAKE  -- COIN

Logic will now allow you to find the fake coin based on the three results. Bearing in mind we don't know whether the fake coin is lighter or heavier.

For instance, if the results were left down, balanced, left down, we could work out which coin is fake in the following way:

From the middle weighing we know that the coins METOFIND are all normal. So one of the coins ACKL is fake. Therefore looking at these coins one at a time in the other two weighings, we can see that:

A - appears on the left twice and could be fake.
C - appears only once, therefore can't be fake (otherwise the first weighing would be balanced).
K - appears on opposite sides, so it can't make the left side go down both times.
L - appears only once, therefore can't be fake (otherwise the third weighing would be balanced).

Therefore the only possibility is A, which must be heavier. Any other combination of ups and downs will allow you to use the same logic to find the fake coin.

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