Rearrange the pieces like this: 
and voila--a 12 by 12 rug without the hole!Where did the hole go? Justify your answer.
FYS: Puzzles and Paradoxes--Problem Set 5
Complete exactly three of the following five problems.
1. The Sultan owns 11 fabulous gems--six giant rubies, three tremendous emeralds, and two spectacular diamonds. Upon his death, he left a curious will. It stated that the gems be divided precisely as follows amongst his three daugthers, Rosa, Esmerelda, and Bianca. Rosa, the eldest was to get half the gems. Esmerelda, the middle child, was to get one quarter of the gems, and Bianca, the youngest was to get one sixth of the gems. Rosa was despondent--how could daddy possibly make me cut one of those beautiful rubies in half? Esmeralda was devastated--how could daddy possibly make me cut up one of those lovely emeralds? Bianca was in tears--how could daddy possibly make me cut a piece off of one of those marvelous diamonds? All of them wailed--"Executing daddy's will without cutting up these gems is impossible!" The local wise woman heard the daughters wail and said--"let me consult my crystal ball." She pulled out her crystal ball and peered into its hypnotic depths. After a few minutes she declared, "I have a solution." She promptly took her crystal ball and put it in amongst the gems. "Half goes to the eldest. Half of twelve is six." Rosa promptly took the six rubies. "One quarter goes to the middle child. One quarter of twelve is three." Esmerelda promptly took the three emeralds. "One sixth goes to the youngest, and one sixth of twelve is two." Bianca promptly took the two diamonds. "And this," pocketing the crystal ball, "is mine," said the wise woman. Satisfied, but mystified, all three daughters wondered where their reasoning to the claim that executing the Sultan's will without cutting up the gems was impossible had gone wrong.
Were they wrong or not? Justify your answer.
2. Of four men and four women, each loves exactly one person and is loved by exactly one person. No one both loves and hates anyone. David does not love either Brad or Charles, but loves the person who loves George. Hortence hates David and is hated by every man, but does not love Francesca. Eloise loves a man, but not George who loves the person who loves the person who loves David. Francesca is loved by the person who is loved by the person who is loved by Abigail. Brad is loved by a man other than Charles, loves a woman, and hates every other woman.
Is it possible to determine who loves Hortence? Justify your answer.
3. A man is sitting in bed unable to fall asleep. He makes a phone call, says nothing, hangs up the phone, and then goes to sleep. Why?
Write a best explanation solution to this situation. Justify that your solution is indeed a best explanation solution.
4. Suppose that psychologists have discovered that witnesses to a single-car accidenct are 87% likely to be able to report the color of the car correctly. Now, suppose that 94% of the taxis in Altoona are yellow, and the remaining six percent are orange. A taxi dents a light-pole and speeds away; a witness reports to the police that the taxi was orange.
The police reason as follows: since witnesses are 87% likely to report the color of cars in an accident correctly and since the witness reports that the taxi involved in the accident is orange, there is an 87% chance the taxi involved in the accident is orange, so we should be on the lookout for a dented orange taxi.
Is the police's reasoning correct or not? Justify your answer.
5.[Bonus Point Problem] Mr M., the magician, had an accident with one of his candle tricks and managed to burn a hole in his 12 foot by 12 foot carpet. So he brings his rug to Mr. R, the rug dealer and asks him to cut and sew the rug to make the hole disappear, but keep the rug 12 feet by 12 feet. "That's impossible," says R, "even for a world famous magician." Mr. M replies: "Impossible, smimpossible. Just cut the rug into pieces like this:
Rearrange the pieces like this: and voila--a 12 by 12 rug without the hole!
Where did the hole go? Justify your answer.