Pre RMO or RMO Solutions: Homi Bhabha Science Foundation

Problem:

Reference: Problem Primer for the Olympiad by C. R. Pranesachar, et al. Prism Books

Amazon India link: https://www.amazon.in/Problem-Primer-Olympiad-2Ed-Pranesachar/dp/8172862059/ref=sr_1_2?s=books&ie=UTF8&qid=1518894790&sr=1-2&keywords=problem+primer+for+the+olympiad

Five men A, B, C, D, E are wearing caps of black or white colour without each knowing the colour of his own cap. It is known that a man wearing a black cap always speaks the truth while a man wearing a white cap always lies. If they make the following statements, find the colour of the cap worn by each of them:

A: I see three black and one white cap.

B: I see four white caps.

C: I see one black and three white caps.

D: I see four black caps.

Solution:

Suppose E is wearing a white cap:

Then, D is lying and hence must be wearing a white cap. Since D and E both have white caps, A is lying and hence, he must be wearing white cap. If C is speaking truth, then C must be wearing a black cap and B must be wearing a black cap as observed by C. But, then B must observe a black cap on C. Hence, B must be lying. This implies that B is wearing a white cap which is a contradiction to C’s statement.

On the other hand, if C is lying, then C must be wearing a white cap. Thus, A, C, D and E are wearing white caps which makes B’s statement true. But, then B must be wearing a black cap and this makes C statement correct.

Thus, E must be wearing a black cap. This implies that B is lying and hence, must be having a white cap. But, then D is lying and hence, must be having a white cap since B and D have white caps. A is not saying the truth. Hence, A must be wearing a white cap. These together imply that C is truthful. Hence, C must be wearing a black cap. Thus, we have the following distribution:

A: white cap; B: white cap; C: black cap; D: white cap; E: black cap.

Hope you enjoyed it! There can be some other approaches too starting with some other assumption(s).

Nalin Pithwa.

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