FYS: Puzzles and Paradoxes--Problem Set 2
Complete exactly three of the following five problems.
1. When it comes to making logical deductions, David's three friends, Arthur, Betty, and Charles never make mistakes. One day David showed his three friends seven stamps: two violet ones, two orange ones, and three teal ones. Then he blindfolded them, and pasted one of the seven stamps on each of their foreheads; the remaining four stamps were removed from the room. When the blindfolds were lifted and each had a chance to examine the others, David asked Arthur, "Do you know one color that is definitely not the color of your stamp?" Arthur replied, "No." Betty frowned when she heard this and when David turned to Betty and asked the same question, she also replied "No". No one can see his or her own stamp.
Is it possible, from this information, to deduce the color of either Arthur's, Betty's, or Charles'stamp? Justify your answer.
2. Definition: A question is problematic iff it is not definitively answerable because at least one of the following obtains: (a) asking the question requires a false presupposition; (b) the question is ambiguous; it is asking two (or more ) questions with different answers; (c) no conventional stipulation has been put in place for answering the question.
Consider the following question: Is it possible for an omnipotent God to create a genuine U.S. currency twenty dollar bill?
Is the question problematic or not? If not, explain and justify why not (and give the answer). If it is, explain and justify why it is.
3. Alice has decided to throw a small dinner party. She invites her father's brother-in-law, her brother's father-in-law, her father-in-law's brother, and her brother-in-law's father.
How many guests does Alice invite, if the number is to be the absolute minimum? Justify your answer.
4. You are the most senior pirate of a group of five pirates which has just captured a treasure of 100 gold pieces. To divide up the treasure, the group uses the following scheme. The highest ranking pirate, (that's you!) makes a proposal on how the treasure should be divided amongst them. Then all the pirates (including the one who made the proposal) vote on the proposal. If the proposal gets 50% or more of the vote, it passes and the treasure is divided up according to the proposal. Otherwise, the other pirates kill the one who made the proposal, and repeat the process with the next most senior pirate. Each pirate will vote to maximize the number of gold pieces he or she receives. However, any pirate will also vote against a current proposal if they believe they would get at least the same number of pieces on a future proposal.
What would you propose to get the most gold possible? (And to stay alive, of course!) Justify your answer.
5.[Bonus Point Problem] You have infiltrated the Secret Convention of High Logicians. But to thwart just such infilatrations, the Master Logician places a dot on each attendee's head, such that everyone else can see it but the person themselves can not. There are many, many different colors of dots. At the convention all attendees sit in a circle in Convention Hall A, and the Master instructs that a bell is to be rung every five minutes: at the moment when an attendee knows the color of the dot on his or her own forehead, the attendee is to leave at the next bell and move to Convention Hall B. Anyone who leaves at the wrong bell is clearly not a true High Logician, but an evil infiltrator, and will be thrown out of the Convention post haste. Once this process has been completed, the convention will proceed in Conventional Hall B. At the beginning of the convention the Master reassures the group by stating that the puzzle is solvable for High Logicians.
Is it possible for you to determine at which bell you ought to leave for Convention Hall B? Justify your answer.