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12 coins

Problem:
There are 12 identical coins. There might be one that is a counterfeit, which is either heavier or lighter than the rest. Can you identify the counterfeit (if any) using a balance at most 3 times?

Solution:
I think this is one of the most difficult variations of the coins weighing problem.

The main idea is to "mark" the coins after weighing them.

Step 1. Divide the coins into groups of 4. Weigh the first two. If they have the same weight, go to step 4. Otherwise, mark the coins belonging to the heavier group 'H', lighter group 'L', and the unweighted ones 'S'. We know that the counterfeit is either in L or H.

Step 2.
Now we have 4 Hs, 4 Ls, and 4 Ss. We now form 3 groups: HHL, LLH, and HLS.

Step 3.
Weigh HHL against HLS.

Case 3.1: HHL is lighter than HLS.
Then either the L in HHL or the H in HLS is the counterfeit. Simply weigh one of them against S and conclude accordingly.

Case 3.2: HHL is heavier than HLS.
Then the counterfeit is one of the 2 Hs in HHL or the L in HLS. To find out which, weigh the 2 Hs against each other. If they are the same weight, L is the counterfeit. Otherwise, the heavier H is the counterfeit.

Case 3.3: HHL is the same weight as HLS.
Then counterfeit is in LLH. Weight an L against the other L. Conclude accordingly.

Step 4.
Here, the counterfeit must be within the 4 unweighted coins. Weigh 3 of them against any 3 coins from the ones weighted in step 1.

If they are the same weight, then the last unweighted coin is the counterfeit.

Otherwise, you now know if the counterfeit is heavier or lighter. Weigh once more and conclude accordingly.

QED

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