Problem:
Five cards are lying on a table as shown.
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Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?
Answer Choices:
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B.
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Solution:
If Jane is wrong, then there is a card with a vowel on one side and an odd number on the other side. Such a card cannot have a consonant or an even number on either side. Thus the only card which could prove Jane wrong is the one with a "" on one side. The other side would have to be a vowel.
The easiest way for Mary to show Jane was wrong is for her to turn over a card showing a vowel, but there is no such card. But if Jane were correct, then so is the contrapositive: "If an odd number is on one side of a card, then a consonant is on the other side." Thus, Mary showed Jane was wrong by turning over the card marked with a .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions