Question

Each of three friends knows whether the other two have passed or failed in an examination, but does not know his own result. The teacher comes and says, "At least one has failed." If all three still do not know their own results, which of the following is true?

• A. One student has failed.
• B. Two students have failed.
• C. Two or more students have failed.
• D. All three have failed.

Hint:

In logic puzzles where players see others' results but not their own, if an "at least one" clue does not lead to an immediate deduction, it often means the minimum failure count is higher than one.

Answer 1

The Correct Option is C

Step 1: Understanding the information each student has. Each student can see the results of the other two, but does not know his own. The teacher's announcement — "At least one has failed" — is public knowledge. Step 2: Considering the case of exactly one failure. - If there were exactly one failure, the two students who see the failed person would know immediately that they have passed. - This would allow them to deduce their result instantly after the teacher's statement. - But the problem states that all three still do not know their results. - Therefore, the case of exactly one failure is impossible. Step 3: Considering the case of all three failing. - If all three failed, each student would see two failures. - In that case, they could not conclude whether they themselves passed or failed without knowing the teacher's statement. - However, after hearing "at least one has failed", this still gives no new information, so uncertainty remains possible. Step 4: Considering the case of exactly two failures. - Suppose two have failed and one has passed. The one who passed sees two failures and cannot determine his own result immediately. - The ones who failed see one pass and one fail; without further information, they cannot deduce their own status immediately either. - Therefore, this situation is also consistent with the problem statement. Step 5: Conclusion. Since exactly one failure is ruled out, the only consistent possibilities are: - Exactly two failures, or - Exactly three failures. Thus, the definite conclusion is that two or more students have failed.