FYS: Puzzles and Paradoxes--Problem Set 1
Complete exactly three of the following five problems.
1. On the island Bivale, there are only two kinds of people, those who always tell the truth and those who always lie. One day, the first of many supplicants to the Supreme Ruler of Bivale announces "Every supplicant but the first and second is a liar." The second supplicant then says, "Every supplicant but the first supplicant is a truth-teller."
Is it possible, based solely on the information given, to determine what kind of person both the first and second supplicant is? Justify your answer.
2. In the land of Trivale there are three types of natives, Knights, Knaves, and Mediators. Knights always tell the truth; Knaves always lie; and Mediators can lie or tell the truth as they please. One day, Jethro, a visitor to Trivale, comes across three natives, Llwellyn, Obadiah, and Priscilla. Though Jethro does not know who is what, he does know that one is a Knight, another is a Knave, and the remaining individual is a Mediator. Luckily for Jethro, the three make the following statements:
Llwellyn: Obadiah is a Knight or Priscilla is a Knave.
Obadiah: I am telling the truth and I am not the Mediator
Priscilla: If I am not a Knight, then I am a Knave.
Is the Mediator lying or telling the truth? Justify your answer.
3. Imagine the following equation is constructed using ten matchsticks: XI + I = X (two each for each X, one each for each I, two for the + and two for the =). Now suppose that adding a match costs $5, removing a match costs $3, and moving a match somewhere else in the equation costs $7. What is the minimum you would need to pay to make the equation correct? Justify your answer.
4. "the first thing that came before him was a question that was submitted to him by a stranger, in the presence of the majordomo and the other attendants, and it was in these words: "Senor, a large river separated two districts of one and the same lordship- will your worship please to pay attention, for the case is an important and a rather knotty one? Well then, on this river there was a bridge, and at one end of it a gallows, and a sort of tribunal, where four judges commonly sat to administer the law which the lord of river, bridge and the lordship had enacted, and which was to this effect, 'If anyone crosses by this bridge from one side to the other he shall declare on oath where he is going to and with what object; and if he swears truly, he shall be allowed to pass, but if falsely, he shall be put to death for it by hanging on the gallows erected there, without any remission.' Though the law and its severe penalty were known, many persons crossed, but in their declarations it was easy to see at once they were telling the truth, and the judges let them pass free. It happened, however, that one man, when they came to take his declaration, swore and said that by the oath he took he was going to die upon that gallows that stood there, and nothing else. The judges held a consultation over the oath, and they said, 'If we let this man pass free he has sworn falsely, and by the law he ought to die; but if we hang him, as he swore he was going to die on that gallows, and therefore swore the truth, by the same law he ought to go free.' It is asked of your worship, senor governor, what are the judges to do with this man? For they are still in doubt and perplexity; and having heard of your worship's acute and exalted intellect, they have sent me to entreat your worship on their behalf to give your opinion on this very intricate and puzzling case."[Don Quixote, Book 2 Chapter 51]
The judges send the man to Sancho (Don Quixote's squire, who has been made governor of a very strange island) because they judge the man's sworn statement as paradoxical. Are they correct? Justify your answer.
5.[Bonus Point Problem] Professor McSnurd is brilliant, but odd. On Mondays and Tuesdays he knows which statements are true and which are false. On Wednesdays and Thursdays, however, he is crazy and takes all true statements to be false and all false statements to be true. To make matters worse, on Mondays and Wednesdays, McSnurd always tells the truth (as he sees it), while on Tuesdays and Thursdays, he always lies (as he sees it). On the other three days McSnurd is nowhere to be found and never says a word.
One day a student asked McSnurd "Is today Thursday?" McSnurd answered with a 'yes' or with a 'no' and the student still did not know what day it was, so the student asked, "Do you believe today is Thursday?" Upon hearing McSnurd's answer, again a 'yes' or a 'no', the student then knew exactly what day it was.
Is there enough information given in the problem for you to determine what day it was? Justify your answer.