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Interview prep / Brainteaser/Logic

Identifying five bags of 9, 10 or 11 g coins at once

Question

Five bags each hold many coins, and within a bag all coins weigh the same: either 99, 1010 or 1111 grams. Different bags may have different weights. What is the fewest number of readings on an exact digital scale needed to determine the coin weight of every bag?

Short answer

One weighing. Take 1,3,9,27,811,3,9,27,81 coins from bags 11 to 55. The reading minus 12101210 equals ∑idi3i−1\sum_i d_i 3^{i-1} with each di∈{−1,0,1}d_i\in\{-1,0,1\}, and balanced ternary representations are unique, so the deviation decodes every bag's weight.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 2.6 The Pigeon Hole Principle, 'Counterfeit coins II', p.21; question and answer rewritten by SelfMadeQuant.com.

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