Problem 24 · 2024 Math Kangaroo
Stretch
Logic & Word Problems
casework
Carl tells the truth one day, lies the next, tells the truth again the day after, and so on. On one day he made exactly four of the following five statements. Which statement can he not have made on that day?
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Answer: A — 2024 is divisible by 11.
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Hint 1 of 2
First decide whether the day is a truth-day or a lie-day, then every statement he made must match that.
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Hint 2 of 2
Two of the statements have a truth value you can settle right away no matter what day it is.
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Approach: fix whether it is a truth-day or lie-day, then test the fixed statements
- Statement A is just a fact: \(2024=11\times184\), so A is always true; statement E ("truth today and truth tomorrow") is impossible because today and tomorrow always have opposite truth-status, so E is always false.
- On a truth-day statements B (tomorrow Saturday) and D (yesterday Wednesday) cannot both be true, so at most three true statements are available; he cannot reach four, ruling out a truth-day.
- So it is a lie-day, where every statement he made must be false; A is true, so A is the statement he could not have made, answer A.
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