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JMM Questions

Set 1:

Hat Strategy

You are one of three prisoners held by a deranged but brilliant statistics professor.  You are told that in a little while the three of you will be put to sleep; when you awake, each of you will be wearing either a white or black hat, where the color of each hat will be chosen by flipping a fair coin (heads is black).  Each of you, knowing only the color of the other two hats but not your own, must then secretly write down either a color or "pass," and at least one of you must not pass. If each of the written colors matches the wearer's hat, then all of you will be set free; otherwise...  What strategy should the three of you agree to use in order to maximize your chance of freedom?

Isle of Blue Eyes

An anthropologist visits a small island where everyone is in daily contact with one another.  The inhabitants, brilliant logicians, have a custom that anyone with blue eyes must leave the island forever at midnight of the day when they learn that they have blue eyes.  However, the inhabitants are compassionate people and would never tell someone that they have blue eyes, and there are no mirrored surfaces on the island.  The anthropologist, as she leaves, expresses to the assembled inhabitants her sadness, as she is sure that someone listening will someday learn that they have blue eyes and have to leave the island.  What happens after she leaves, and when?

Four-Footed Bridge

Four people need to quickly cross a rickety wooden bridge over a deep ravine.  The bridge can only hold two of them at once.  Also, it is a pitch black night and the bridge is missing many floor planks; they have one flashlight that must be carried with whoever is crossing in order to avoid plunging into the ravine.  Furthermore, some of them can move faster than others: Paul can get across in one minute, John takes two minutes, George takes five, and Ringo takes 10.  How quickly can all four get safely across?

Open-and-Shut Case

You live alone on an island, and your friend, who lives on another island, calls to say that he is ill and needs the medicine that you have.  There is one boat that travels between your islands several times each day, but its captain is not trustworthy: she will take anything that is not securely locked in the boat's treasure chest.  This chest has a large clasp, and you have a lock that can be fastened to the loop of this clasp in order to secure the chest.  Your friend also has his own lock that could similarly secure the chest.  However, each of you only has a key to your own lock, and the keys do not fit the other locks.  How can you safely get the medicine to your friend?

Question Time

You find yourself in a room with two doors and two guards.  You know that one of the guards always tells the truth and that the other always lies, but you do not know which is which.  You also know that one door leads to freedom, the other to, well, something bad.  You are allowed to ask one guard one question, after which you will be allowed to go through one door.  What question should you ask?

Burning to Know

You are given two length of string and one match.  Each string burns for exactly one minute, but they might not burn at a uniform rate (each might burn faster in some portions of the length, slower in others).  Use these strings to measure a duration of 45 seconds.

Set 2:

Check the Boxes

You are one of four prisoners held by a deranged but brilliant statistics professor.  To survive execution, each of you must pass a test:  each distinctly-named prisoner has a box with his name on it, and each box contains a piece of paper with one of the four names written on it.  One by one, each of you must find his own paper by looking in at most two boxes.  You are not allowed to converse in between rounds.  If everybody passes, you are all set free; otherwise... What strategy should the four of you agree to use in order to maximize your chance of freedom?

Half and Half

Suppose you are given a rectangle, which a smaller rectangle completely contained inside of it.  The smaller rectangle need not have any sides parallel to any of the sides of the larger.  Devise a simple procedure for cutting the area between the two rectangles perfectly in half.  Your procedure should not call for the calculation of any area.

Getting Warmer?

The following statement is true:  At any point in time, there exist two points on the surface of the moon that are opposite (that is, the line directly connecting them passes through the center of the moon) that have the exact same temperature.  Why is this true?

Make it a Big One

Given ten seconds to write down a large number (using standard math notation and/or English words), what would be your approach to maximize the number written down?  Your answer should be precise enough for any reasonable modern mathematician to exactly compute, by consulting only your card, and, if necessary, the published literature.

Winning Numbers

The presidential elections are to be held in Anchuria. Of 20,000,000 voters only 1 percent (i.e. the regular army) support the current president Wobushko. He wants to be re-elected in a democratic way, which means the following. All voters are split into n1 groups, all of equal size. Then each group can be split into n2 smaller sub-groups of equal size, where n2 is the same for all groups. Then each subgroup is split into n3 equal sub-sub-groups, and so on. Each (sub)i-group chooses by majority rule one representative to represent it at level i-1, and so on. (If there is a tie, the opposition wins.) Can Wobushko organize the groups and distribute his supporters so that he wins the elections?

Position that Pays

Two players alternatively erase some 9 numbers from the sequence 1,2,...,101 until only two remain. The player that starts wins x-54 dollars from the player that plays second. Here x is the difference between the remaining two numbers. Would you rather be the first or the second player?

Answers are on the next page >>.