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Divide the 11 coins into three trios and one pair. Let the trios be A,B,C. Weigh A against B. Then weigh A against C. From this we can determine whether the odd coin lies in any of the trio or not.
Case 1: the odd coin does not lie in any trio.
This happens when both weighings are balanced. Then the odd coin must be in the pair. Take any one coin from any trio (which are not odd), then weigh against any coin from the pair. If it shows balance, then the other coin from the pair is odd. If it is not balanced then the one on the balance (from the pair) is odd. Then we are done by three weighings.
Case 2: the odd one lies in the trio.
If this is the case, then we can deduce which trio contains the odd coin, since, we have only three possibilities:
(i) A equal B, A unequal C => the odd coin is in C
(ii) A unequal B, A equal C => the odd coin is in B
(iii) A unequal B, A unequal C => the odd coin is in A
And also, we can easily check that we can determine whether the odd coin is lighter/heavier. Now take the trio where the odd coin is supposed to be. Then weigh any two of them: if it is equal then the third one is odd; if not then the lighter/heavier of them (depending on our observation earlier) is odd. We are done with three weighings. |
