Problem 20 · 2017 Math Kangaroo
Hard
Logic & Word Problems
casework
Lilli tries to be a well-behaved kangaroo, but she is having just too much fun not to lie every now and then. So every third statement of hers is a lie and the rest are true; sometimes she starts with a lie and sometimes with one or two true statements. Lilli thinks of a two-digit number and says to her friend:
1: “One digit of the number is a 2.”
2: “The number is greater than 50.”
3: “It is an even number.”
4: “The number is less than 30.”
5: “The number is divisible by 3.”
6: “One digit of the number is a 7.”
What is the sum of the digits of the number Lilli is thinking of?
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Answer: D — 15
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Hint 1 of 2
Exactly one statement in each block of three is a lie, so the lies sit at positions {1,4}, {2,5}, or {3,6}.
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Hint 2 of 2
Test each lie-pattern: only one makes all the 'true' statements consistent for a real two-digit number.
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Approach: casework on which statements are the lies, then find the number
- The two lies are at positions {1,4}, {2,5}, or {3,6}. The first two cases give contradictions (e.g. '>50' and '<30' both true).
- With lies at {1,4}: the number is even, greater than 50, divisible by 3, contains a 7, contains no 2, and is at least 30.
- That number is 78 (even, >50, 7+8=15 divisible by 3, has a 7, no 2).
- Its digit sum is 7 + 8 = 15.
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