A game show asks four contestants to list five historical events, labeled A, B, C, D and E, in chronological order. They receive one point for each event that they position correctly in the order—so if the correct order is A B C D E and a contestant guesses E B A D C, they will get two points for B and D being placed correctly. Here are the contestants’ guesses and scores:
What is the correct order?
The correct order is B D C E A.
The first two guesses differ only by swapping D and E; A, B and C stay fixed. The second guess earned two points. They can’t both come from E and D because then the first guess would have earned zero points. They also can’t both come from A, B, and C because then the first guess would have earned at least two points. So for Contestant 2’s answer, one of A B C _ _ is correct and one of _ _ _ E D is correct.
The same logic applies to the first and fourth guesses. These only differ in swapping A and B while C, D and E stay fixed. By the same reasoning, we conclude that one of _ _ C D E is correct and one of B A _ _ _ is correct. Furthermore, we know the order of A B in Contestant 1’s answer is wrong because otherwise the first guess would have earned points from A or B and from C, D or E, but the first guess only earned one point.
Because one of A B C _ _ is correct but A B _ _ _ are both wrong, C must be correct in the middle position.
The third guess, E A C B D, earned one point, which must have come from the C, so all of the other positions are wrong. We established earlier that one letter from Contestant 2’s E D placement is correct. It now can’t be D because Contestant 3’s answer shows D is wrong in the final position. Similarly, we learned that one letter from Contestant 4’s B A placement is correct, and we know it can’t be A because Contestant 3’s answer shows A is wrong in the second position. So B is placed first, and E is placed fourth. Only A and D remain, and only one placement avoids contradicting Contestant 3’s answer. The correct ranking is therefore B D C E A.
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