Problem 22 · 1986 AJHSME
Stretch
Logic & Word Problems
implication-chain
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
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Answer: C — Carlos, Diana.
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Hint 1 of 3
Picture the three statements as falling dominoes pointing one way: Alan → Beth → Carlos → Diana. If a domino early in the line falls (gets an A), every domino after it must fall too. So who can afford to get an A without toppling too many?
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Hint 2 of 3
Count the dominoes that fall from each starting point. Alan's A topples 4 in all; Beth's topples 3; only starting near the *end* of the chain keeps the total at exactly 2.
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Hint 3 of 3
The arrows only point forward, so getting an A never forces anyone *earlier* — that's why the two A-getters must be the last links.
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Approach: follow the forward-only domino chain
- Each 'if I get an A, then the next person does' is a one-way arrow: Alan → Beth → Carlos → Diana. An A anywhere knocks down everyone *after* it, but never anyone before.
- Test from the front. If Alan gets an A, the chain forces Beth, Carlos, and Diana too — that's 4 A's, too many. If Beth gets an A, it forces Carlos and Diana — 3 A's, still too many. So neither Alan nor Beth can be one of the two.
- That leaves the tail end. If Carlos gets an A, only Diana is forced — exactly 2 A's, and nobody else is dragged in: Carlos and Diana.
- Why start from the front: because the arrows point forward, the *earlier* a person is, the more A's they trigger. To keep the count smallest, the A's must sit as far down the chain as possible — a handy rule for any 'if-then' chain problem.
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