Problem 30 · 2025 Math Kangaroo
Stretch
Logic & Word Problems
caseworkwork-backward
Mike has three bags. Each bag contains three balls. On one bag there is a sign saying “1 white, 2 black”, on the second a sign saying “2 white, 1 black” and on the third a sign saying “3 white”. However, the signs have been swapped so that none of them is correct now. On each turn, Mike chooses a bag that still contains balls, draws one blindly and places it visibly next to the bag. What is the minimum number of balls that he has to draw to know for sure which sign should have been on which bag?
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Answer: C — 2
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Hint 1 of 3
Since every label is wrong, the labels form a derangement of three—so naming one bag's true contents forces the other two.
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Hint 2 of 3
Pick the bag that gives the most information per draw, and ask what a single draw can fail to settle.
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Hint 3 of 3
Test whether one draw can ever be ambiguous, then whether two draws always resolve it.
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Approach: use that the labels are a derangement, then bound the draws
- Because no label is correct, the three labels are a derangement of three, so identifying any one bag's true contents determines all three.
- Draw from the bag labelled "2 white, 1 black": its real contents are either "1 white, 2 black" or "3 white," so one black ball settles it—but a first white ball is still ambiguous, so one draw is not enough.
- A second draw from that same bag always distinguishes the two cases, so 2 draws suffice and are necessary, choice (C).
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