Assignment 1: Logic Application
Andy, Belle, Carol, and I are playing the game Guess Your Card. In the game, each person draws three cards without looking from a stack of cards containing contain multiple cards ranging in denomination from one to nine. Each person then places the cards on his or her forehead so that all of the other players can see the others’ cards, but cannot see their own. There is also a stack of questions that each person draws from in turn. These questions help the players deduct the identities of their own cards.
We have shuffled the deck and each player has drawn three cards and placed them on their own forehead. Andy has drawn 1, 5, and 7; Belle has drawn 5, 4, and 7; and Carol has drawn 2, 4, and 6. Obviously I cannot see my own cards. Andy draws the first question, which asks, “Do you see two or more players whose cards sum to the same value?” To which he answers, “Yes.” Belle’s turn is next. Her card asks, “Of the five odd numbers, how many different ones do you see?” She responds, “All of them.”
With these two questions, I am able to deduce which cards I have. After Andy drew the first question, “Do you see two or more players whose cards sum to the same value?” I added up Belle’s and Carol’s cards to see if theirs sum to the same total. Belle’s cards (5,4,7) add up to 16. Carol’s cards (2,4,6) add up to 12. Since Belle’s and Carol’s cards do not add up to the same amount, I can conclude that my cards add up to either 16 or 12. The next question regarding how many odd numbered cards Belle sees, indicates that I have the two remaining odd numbers, as the only odd numbered cards visible to me are 1,5, and 7. At this point, I know that I have 3 and 9. Since Belle’s cards add up to 16 and Carol’s to 12, and my 3 and 9 cards add up to 12, I know that the unknown card must be 4, which added to 3 and 9, makes 16.
Andy is able to deduce his cards based only on Belle’s question, “Of the five odd numbers, how many different ones do you see?” Andy can see that I have 3, 9, and 4, Carol has 2, 4, and 6, and Belle has 5, 4, and 7, which she cannot see. This indicates that the only odd numbered cards both she and Andy can see are my 3 and 9, leaving 1, 5, and 7. Andy deducts that he must have the 1, 5, and 7.
With the two given questions, it would not yet be possible for Belle or Carol to deduce the identities of their cards. Belle knows that her cards must not sum to 16, as both Carol’s and my cards sum to 16. Carol knows that her cards must sum to either 16 or 12, because Belle’s cards sum to 12 and my cards sum to 16. However, neither Belle nor Carol has sufficient information to deduce what their own cards are.
This game uses logic, deductive reasoning, set theory, elimination, and basic knowledge of math. In order for each player to guess their own cards, they must pay attention to the answers the other players give to each question and use the information to eliminate possibilities. The right question can give away the answer, as in Andy’s case, or a series of questions can lead to a conclusion, as in my case.