Fewest coins making 60 cents ambiguously

Question

Brian's pocket holds exactly 60 cents, made up only of nickels, dimes and quarters. If you are told he has $N$ coins, for which smallest $N$ is it impossible to deduce how many of each coin he has?

Short answer

6. Listing all mixes worth $60$ cents, the coin counts $3,4,5$ each arise from exactly one mix, but $6$ coins can be $1$ quarter, $2$ dimes and $3$ nickels or $6$ dimes. The ambiguity comes from swapping one quarter plus three nickels for four dimes.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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