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Jeffrey Self Professor Mune Lokesh Math 104 Algebra With Applications 19 June 2013

In my scenario I am sitting around the kitchen table with three friends and we decide to play the game Guess Your Card. My friends explain to me that in this game each of us in turn will blindly draw three cards from a partial deck of cards. This deck has had all cards above 9 removed, and for this game’s purposes aces are only equal to 1. Without looking at my cards I am to place them facing out on my forehead so the other players can clearly see the cards. Each of the other players is to do the same. The object of the game is to guess which cards you have on your own head using logic to solve the problem. Players draw question cards that reveal information about player cards, and the first to guess their cards based on the information revealed wins the game. My brother Andy has the cards 1, 5, and 7 showing on his forehead. His wife Belle has 5, 4, and 7. My girlfriend Carol has the cards 2, 4, and 6. Andy is selected to draw the first question card, and it asks if he sees two or more players who’s cards sum to the same value. Andy answers affirmatively. With this information I know that I must have cards that equal either 12 or 16. I know this because Belle’s cards sum to 16 and Carol’s cards sum to 12. Since Andy sees two of us with cars of the same value I am able to confirm that my sum must match either Belle’s or Carol’s. At this point none of us are able to conclude anything else about our cards, so Belle draws the next question card. Belle’s question reads “Of the five odd numbers [1, 3, 5, 7, and 9] how many different odd numbers do you see?” Belle answers that she sees all five the odd numbers. I can know that Belle can see the 1, 5, and 7 that Andy is showing and that Carol has no odd numbers showing at all. This means that I must have both the number 3 card and the number 9 card on my own forehead. With this information I can now solve the

problem. I know that I have a 3 and a 9. From the previous question I also know that my cards must sum up to either 12 or 16. Since 3 and 9 equal 12 and my third card cannot equal zero I know that my cards must total 16. The final step is easy! I simply deduct 12 from 16 and I know that my last card must be a 4. My cards are 3, 9, and 4. I draw a breath to win the game, but Andy beats me to it, blurting out that he knows he has a 1, 5, and a 7. After thinking about it for a few moments I realize that Andy’s problem was an easier one to solve than my own. When Belle was asked how many odd cards she saw and she answered all of them that immediately gave Andy his answer. Andy was able to see that I only had 3 and 9 and that Carol had no odd cards at all. If Belle was able to see all five odd cards that had to mean that he had the remaining three odd numbers. When we first started to play the game it seemed as if it was simply going to be a matter of blindly guessing three cards out of the possible twenty-seven cards available. As the game progressed and small bits of information was revealed I was able to use logic to determine that I was wearing the 3, 9, and 4 cards on my head.

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