Chapter 7: Problem 52

While three wise men are asleep under a tree a mischievous boy paints their foreheads red. Later they all wake up at the same time and all three start laughing. After several minutes suddenly one stops. Why did he stop?

Short Answer

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Answer: One of the wise men stopped laughing because he deduced that his own forehead must also be painted red, based on the observations that the other two wise men didn't stop laughing and taking into account the possible scenarios.

Step by step solution

Initial Observations

Since all three wise men have their foreheads painted red, without any verbal communication, they can observe that the other two have red foreheads. They also know that there are three wise men, and thus begin to think about the possible scenarios.

Analyzing Scenario 1

In this scenario, let's assume that only one of the three wise men has a red-painted forehead. If this were true, the other two wise men would see one with a red forehead, and one without. The one not having any paint on his forehead would immediately know that he himself is not painted, and thus stop laughing. However, since all are laughing, this scenario cannot be true.

Analyzing Scenario 2

In this scenario, let's assume that two of the three wise men have red-painted foreheads. If this were true, the two with red foreheads would each see one other wise man with a red forehead, and one without. Now, the one without a red forehead should have stopped laughing in the first minute (as mentioned in Step 2), but he didn't. This would make both the wise men with red foreheads realize that their foreheads are also painted red. Consequently, they would stop laughing after the second minute. Again, since all are still laughing after several minutes, this scenario cannot be true.

Analyzing Scenario 3

In this scenario, all three wise men have red-painted foreheads. Each wise man sees two others with red foreheads. Since Scenario 2 (two wise men with red-painted foreheads) could not happen, everyone would deduce that all three must have red-painted foreheads. By realizing that their own foreheads must be painted red as well, one of them (it could be any one of them) stops laughing, putting an end to the situation. In conclusion, the wise man who stopped laughing did so because he deduced that his own forehead must also be painted red, based on the observations that the other two wise men didn't stop laughing and taking into account the possible scenarios.

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While three wise men are asleep under a tree a mischievous boy paints their foreheads red. Later they all wake up at the same time and all three start laughing. After several minutes suddenly one stops. Why did he stop?

Short Answer

Step by step solution

Initial Observations

Analyzing Scenario 1

Analyzing Scenario 2

Analyzing Scenario 3

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