Three Mislabeled Jars

Question

Three jars are labeled "Apples", "Oranges", and "Mixed", but every label is wrong. You may draw one fruit from one jar (without looking inside). What is the minimum number of draws needed to relabel all three jars correctly, and how?

Accepted Answer

One draw suffices. Draw from the jar labeled "Mixed". Since all labels are wrong, that jar is not mixed, it's pure apples or pure oranges, so the single fruit you draw identifies it completely. Say you draw an apple → that jar is Apples. Now the jar labeled "Oranges" can't be oranges (wrong label) and can't be apples (taken) → it's Mixed. The remaining jar (labeled "Apples") is Oranges. The general lesson interviewers want articulated: constraints are information. "All labels are wrong" eliminates so many configurations (only 2 of the 6 permutations have zero fixed points... and the draw separates those two) that one bit of new data resolves everything. Quantify it: you need to distinguish 2 possible worlds, and one draw from the mislabeled-Mixed jar yields exactly 1 bit.

Three Mislabeled Jars

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