Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given cards out of a set of cards numbered The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- Oscar-- Aditi-- Tyrone-- Kim-- Which of the following statements is true?
- A)
- B)
- C)
- D)
- E)
Answer
C
Key insight
Start with the most constrained scores: Oscar's 4 forces {1,3}, then Aditi's 7 forces {2,5}, then Ravon's 11 forces {4,7}.
Solution
Attack the smallest score first, since it has the fewest options.
- Oscar has : two distinct cards summing to must be .
- Aditi has : the pairs are . Cards and are gone, so Aditi has .
- Ravon has : the pairs are . Cards are gone, so Ravon has .
- Tyrone has from the remaining : only works ( is Kim's).
- Kim has , summing to . Consistent.
Ravon holds card , so statement (C) is true; the others are false (Aditi has and , Tyrone has and ).
The answer is
Why this works
Extreme scores have few representations as a sum of two distinct cards, and each forced assignment removes cards from later players' options. Processing from the most constrained player outward turns a search into a chain of forced steps. The full assignment is unique, so exactly one statement can be true.
The trap
Assigning Aditi 3 + 4 = 7 before noticing that Oscar's score of 4 already requires cards 1 and 3.
Common mistakes
- Assigning Aditi 3 + 4 = 7 before noticing that Oscar's score of 4 already requires cards 1 and 3.
- Giving Tyrone for without checking that is already Ravon's, which then makes Kim's impossible.
Techniques
Organized listing / direct enumeration · Consider the largest/smallest element or boundary case