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This question already has an answer here:

I hope this has an easy answer!

A batch of $n$ coins has a single fake coin, which is lighter than the rest.

The idea behind this puzzle is:

What is the largest batch of coins that can have the fake coin identified given $k$ weighings?

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marked as duplicate by Bass, Glorfindel, Peregrine Rook, boboquack, Quintec Oct 28 '18 at 1:12

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Re the above comment, it's probably better to link to the source of the duplicates instead of a question which itself is a duplicate. $\endgroup$ – boboquack Oct 28 '18 at 0:57
  • $\begingroup$ Fun fact: Jon even commented on the linked dupe question :P $\endgroup$ – Quintec Oct 28 '18 at 1:14
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With this strategy

For every weighing divide the remaining coins into three equally large groups. Compare two of the groups – if one is lighter, the lighter coin is in that group, and if both have the same weight the lighter coin is in the unweighed group. Continue until the remaining group is just one coin.

We can find the fake coin in $k$ weighings

From at most $3^k$ coins.

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