You have 9 coins. One of the nine is counterfeit. The counterfeit coin can be distinguished by weight - it is heavier than the rest.
Using a balance scale only twice, find the counterfeit coin.
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Split into groups of 3. Call them first3group, second3group, third3group.
Now weigh first3group vs second3group. If first test shows equal weights, the heavy coin is in third3group. Split third3group and weigh 2. Where one is heavier, QED, otherwise, it is the non-weighed one, QED.
Else take the heavier group of 3 and do a similar exercise as described above when left with 3, 1 of which is heavier.