restricted access Chapter 7 The Incredible Shrinking Room
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Chapter 7 The Incredible Shrinking Room Fermat’s Room (La Habitación de Fermat) (2007) is a Spanish movie reminiscent of Cube (1997). Four “mathematicians,” strangers to each other, are invited to a party by a mysterious host who calls himself Fermat. The pretext is the resolving of a great mathematical problem. For the duration of the party, the guests are to use the pseudonyms Galois, Hilbert, Pascal, and Oliva. Away from the party, Galois is a math student who claims to have found a proof of the famous Goldbach conjecture,1 Hilbert is an elderly mathematician, Pascal is an engineer, and all we ever learn of Oliva is that she is very good at chess. The invitation is a trap, and the four are imprisoned in a square room. They are then confronted with mathematical problems. Whenever they fail to solve a puzzle within the specified time, the walls of the room close a distance in on them (an oldie but a goodie). They are not given much time: over one hour, the walls are set to shrink from about 7 meters in length to just 1 meter. Fermat’s Room has many cute mathematical touches: a boat called Pythagoras ; the Kepler conjecture on the densest packing of balls; characters the same age as their mathematician namesakes were when they died (with the notable exception of Hilbert); many examples of the Goldbach conjecture; and of course the puzzles. 1 The Goldbach conjecture is a very old, very famous, and still unsolved mathematical problem. It states that every even integer greater than 2 can be written as the sum of two prime numbers. That is 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, and (so goes the conjecture) so on. The Goldbach conjecture also plays a central role in the murder mystery Inspector Lewis (2006) and in the biopic Chen Jingrun (2001). Jimmy Stewart also plays around with it in No Highway in the Sky (1951). And, in the Futurama episode “The Beast With a Billion Backs” (2008), Farnsworth and Wernstrom collaborate on “another elementary proof.” 97 98 7 The Incredible Shrinking Room 7.1 How Good a Puzzler Are You? In this chapter we’ll focus on the puzzles confronting the trapped mathematicians .2 And we’ll frame it as a challenge for you; see how well you would fare, if you were trapped in Fermat’s deadly room. To begin, we’ll pose a couple of warmups. The first is an easy one, an old chestnut discussed by the four mathematicians while boating on a lake. You’ll find the answer to this and all the puzzles at the end of the chapter. First warmup puzzle: A man has to transport a wolf, a sheep, and a cabbage across a river in a small rowboat. On each trip, the man can only transport one passenger. When the man is not around, the wolf will eat the sheep, and/or the sheep will eat the cabbage. Devise a plan that will move all three passengers safely to the other side of the river. The second warmup puzzle is not so easy. It was the preliminary puzzle sent to the four mathematicians, to test their worthiness of an invitation to the party. Second warmup puzzle: What is the principle behind the ordering 8, 5, 4, 9, 1, 7, 6, 3, 2? Warmup’s over, here are the rules for our game. • There are seven puzzles. • Each puzzle has a time limit of five minutes (in the movie, the times allotted to the puzzles vary). Once the time limit is up, the walls begin moving. Keep track of how much you go over time. If your excess time reaches one hour, we declare you dead. • To compensate for not having others to help you, if you’re stumped by a puzzle you can choose to roll a die. If you roll a 6, we declare you dead. Otherwise, you survive, and can continue with the next puzzle. • If your answer to a puzzle is not correct, then you have to roll the die. • No cheating! Are you ready to go? Good luck! Puzzle 1: A confectioner receives three boxes of candies. One contains only mints, the second contains only chocolates, and the third contains a mixture of the two. The boxes have labels to identify the contents of the boxes. However , the confectioner was informed that all the boxes have changed labels. What is the minimum number of candies...