Thirteen coins when you need not know heavy or light
You have thirteen coins, one of which is fake and either heavier or lighter than a genuine coin; you do not know which. You have a two-pan balance and no reference coin.
It is known that three weighings cannot both identify the fake and determine its direction when there are thirteen coins.
Show that three weighings do suffice if you only need to identify which coin is fake, without necessarily learning whether it is heavy or light.
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Weigh four against four with five aside. The tilted branches work exactly as in the twelve-coin puzzle. The balanced branch leaves five suspects and eight known-genuine coins; use the genuine coins as references, and accept that for one particular coin you may finish without knowing its direction.
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