Logic. Montlake Math Circle. October 22, In The Royal Art Museum of Samhop, a man was looking at portraits in a gallery.
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1 Logic Montlake Math Circle October 22, 2012 Problem 1 In The Royal Art Museum of Samhop, a man was looking at portraits in a gallery. a. As he looked at at a portrait in a great gold frame, a fellow museum goer asked him, Whose picture are you looking at? He replied, Brothers and sisters I have none, but this man s son is my father s son. Whose picture was he looking at? b. The man then proceeded to look at the next portrait and the museum goer asked again, Whose picture are you looking at? He replied, Brothers and sisters I have none, but this man s father is my father s son. Now whose picture was he looking at? 1
2 Problem 2 There are red socks and blue socks placed together in a drawer in a dark room. a. Suppose there are 10 red socks and 10 blue socks. How many socks do you need to take out to guarantee you get at least two socks of the same color? b. Now suppose there is some number of blue socks and the same number of red socks (not necessarily 10). But now suppose we know that the minimum number of socks you need to pick to get at least two socks of the same color is the same as the minimum number of socks you need to pick to get at least two socks of different colors. How many socks are in the drawer? 2
3 Problem 3 Once there was a lone traveler exploring the Kingdom of Samhop. In the Samhop there are Knights and Knaves. Knights can only tell the truth and Knaves can only lie. One day the traveler passed three inhabitants of Samhop Russ, Rusty, and Dale as they walked through the countryside. a. The traveler asked Russ, Are you a knight or a knave? Russ answered, but the traveler could not understand what he said. So then the traveler asked Rusty, What did he (Russ) say? Rusty replied, Russ said that he is a knave. Then Dale said, Don t believe Rusty; he is lying! (i) Did Russ say I am a knave? (ii) What are Rusty and Dale? b. Having figured out Russ, Rusty, and Dale, the traveler moved along. But soon he came upon another trio: Percival, Robin, and Lancabit. The second traveler asked Percival, How many knights are among you? Percival answered but the traveler could not understand what he said. So the traveler asked Robin, What did Percival say? Robin replied, Percival said there is only one knight among us. Then Lancabit said, Don t believe Robin; he is lying! (i) What are Robin and Lancabit? 3
4 Problem 4 Suppose the traveler from Problem 3 comes across two more inhabitants of Samhop, Guinevere and Morgana. The traveler asks Guinevere whether Morgana is a knight. Guinevere gives an answer (yes or no). Then the traveler asks Morgana whether Guinevere is a knight. Morgana gives an answer (yes or no). Did Morgana and Guinevere both give the same answer, or could they have answered differently? 4
5 Problem 5 In the Kingdom of Desbro (one of Samhop s main trading partners), in addition to Kights and Knaves there are also Normals. Normals sometimes tell the truth and sometimes lie. a. The traveler from before comes across three people in a town, Merlin, Gandalf, and Dumbledore. One is a knight, one a knave, and one a normal. They make the following statements: Merlin: I am a normal. Gandalf: That is true. Dumbledore: I am not a normal. What are Merlin, Gandalf, and Dumbledore? b. The traveler comes across two people, Lavender and Luna. Lavender and Luna are each either a knight, a knave, or a normal. They make the following statements: Lavender: Luna is a knight. Luna: Lavender is not a knight. The traveler realizes that at least one of them is telling the truth and is not a knight. does she know? How 5
6 Problem 6 The king of Desbro had a peculiar way of trying prisoners. A prisoner had to choose between two doors. Each had either a lady or a tiger behind it. If prisoner opened a door to a lady, he would be set free and could marry the lady, should he desire. However, if the prisoner opened the door to a tiger, he would be most likely eaten. The king did not like leaving anything to chance, so he put signs on the doors allowing the prisoner to reason which door to choose. One day when the king had a prisoner to try, the king had the rooms prepared, signs placed on the doors, and the prisoner brought before them. The king explained the rules: There could be ladies in both rooms, tigers in both rooms, or a lady in one and a tiger in the other, and either both signs were true or both were false. The signs read: Room 1: A tiger is in this room OR a lady is in the other room. Room 2: A lady is in the other room. Note: A statement containing OR is only false when both parts are false. The prisoner wants to find a lady (nobody likes being eaten!). What should he do? 6
7 Problem 7 Back in the Kingdom of Samhop, the king was trying three of his inhabitants, Octavia, Charlotte, and Ursula, for a crime. It was known that only one of the three committed the crime, only one of the three was a knight, and the knight was the one that committed the crime. They made the following statements: Octavia: I am innocent. Charlotte: That is true. Ursula: Charlotte is not normal. Which one is guilty? 7
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