FYS: Puzzles and Paradoxes--Problem Set 7

Complete exactly three of the following five problems.

1. Two Buckets: Bucket #1 contains a gallon of water and bucket #2 a gallon of wine. Exactly one tablespoon of wine from bucket #2 is poured into bucket #1. Exactly one tablespoon of the resulting mixture from bucket #1 is poured back into bucket #2.
Which of the following is true? Justify your answer.

(i) There is more water in bucket #2 than wine in bucket #1.
(ii) There is more wine in bucket #1 than water in bucket #2.
(iii) There is the exact same amount of water in bucket #2 as wine in bucket #1.

2. River Crossing: A family of sheep comprised of a ram, a ewe and three lambs come to a wide river. The only way across is a small boat which can carry exactly one grown-up or at most two lambs. Mr Wolf, who is a grown-up, will only let the family use the boat to cross the river if he can be guaranteed to get his boat back (trying to send the boat across with no one in it will definitely lose the boat). The family will only use the boat if (a) neither of the youngest two lambs is ever alone on one side of the river and (b) Mr. Wolf is never the only grown-up with any of the lambs.

Is it possible for the family to get across the river such that Mr. Wolf gets his boat back? Justify your answer.

3. The Art Gallery: At 8:00 am you are at the entrance of the Sophia Art Gallery. The gallery is odd in two respects. Firstly, there are no spaces between the paintings--the walls are seemlessly covered with fine art. Secondly, there is one and only one path through the gallery. At 8:01 am, you begin to walk the one and only path through the gallery. Twelve hours later, after numerous breaks to rest and view the various paintings, you finally arrive at the restaurant and hotel located at the point of the path farthest from the main entrance. After a tranquil night and exquisite breakfast, you realize you had better get going or you will be late for a very important date. You start back through the museum at 9:00 am and, after only a brief stop to check out the Vermeers again, you arrive back at the main entrance two hours later.

Is there any painting in the gallery that you were in front of at the exact same time on each day? Justify your answer.

4. Location, Location, Location: The planet Loki is twice the diameter of Earth, but, since it contains fewer heavy metals, has approximately the same mass as Earth. Unfortunately, while Loki has an Earth-normal gravity and atmosphere, it has no stable magnetic pole (which rarely matches with the geographic north pole anyway) and is subject to violent electrical storms that wreak havoc on navigation systems, as the first Terran Planetary Survery mission to Loki has discovered. Prior to the dispatch of a rescue team, the last transmission from the Survey mission was as follows:
Most of supplies lost in earthquake. Have lost our bearings--location unknown. Yesterday went ten kilometers due south, ten kilometers due east, and ten kilometers due north and arrived right back at starting point, but still our location unknown. Send help.

How many points on the surface of Loki satisfy the claims made by the survey team:
(i) Exactly one
(ii) Exactly two
(iii) More than two.

Justify your answer.

5.[Bonus Point Problem] Hell: Zulu. Zulu is trapped in Hell. Though the Devil's former victims, Foxtrot, Golf, Hotel, Lima, ..., are not guaranteed to get out of Hell, given the real numbers the Devil has chosen for each of them, the Devil is bored with picking real numbers. So, for Zulu, he decides on a different game. The Devil will think of a finite set of positive integers. The set can contain duplicates; e.g. the set {1, 1, 2, 2 } is an allowable set. (Once the Devil decides on his finite set for Zulu, the Devil will never change it.) On each day, Zulu gets to guess one finite set of positive integers. If Zulu guesses the Devil's set, then Zulu gets to go free.

Is there a way that Zulu can guarantee his escape from Hell or not? Justify your answer.

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