• # Brain Teasers & Puzzles

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• A building has 16 rooms, arranged in a 4×4 grid.  There is a door between every pair of adjacent rooms (“adjacent” meaning north, south, west, and east, but no diagonals).  Only the room in the northeast corner has a door that leads out of the building.

In the initial configuration, there is one person in each room.  The person in the southwest corner is a psycho killer.  The psycho killer has the following traits:  If he enters a room where there is another person, he immediately kills that person .  But he also cannot stand the site of blood, so he will not enter any room where there is a dead person.

As it happened, from that initial configuration, the psycho killer managed to get out of the building after killing all the other 15 people.  What path did he take?

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• You have two jars.  One contains vinegar, the other oil.  The two jars contain the same amount of their respective fluid.

Take a spoonful of the vinegar and transfer it to the jar of oil.  Then, take a spoonful of liquid from the oil jar and transfer it to the vinegar jar.  Stir.  Now, how does the dilution of vinegar in the vinegar jar compare to the dilution of oil in the oil jar?

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• You’re given a procedure that with a uniform probability distribution outputs random numbers between 0 and 1 (to some sufficiently high degree of precision, with which we need not concern ourselves in this puzzle).  Using a bounded number of calls to this procedure, construct a procedure that with a uniform probability distribution outputs a random point within the unit circle.

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• Initially, you’re somewhere on the surface of the Earth.  You travel one kilometer South, then one kilometer East, then one kilometer North.  You then find yourself back at the initial position.  Describe all initial locations from which this is possible.

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• The games played in the soccer world championship form a binary tree, where only the winner of each game moves up the tree (ignoring the initial games, where the teams are placed into groups of 4, 2 of which of which go onto play in the tree of games I just described).  Assuming that the teams can be totally ordered in terms of how good they are, the winner of the championship will indeed be the best of all of the teams.  However, the second best team does not necessarily get a second place in the championship.  How many additional games need to be played in order to determine the second best team?

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• You’re given a 3x3x3 cube of cheese and a knife.  How many straight cuts with the knife do you need in order to divide the cheese up into 27 1x1x1 cubes?

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• You have 12 coins, 11 of which are the same weight and one counterfeit coin which has a different weight from the others.  You have a balance that in each weighing tells you whether the two sides are of equal weight, or which side weighs more.  How many weighings do you need to determine:  which is the counterfeit coin, and whether it weighs more or less than the other coins.  How?

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• A room has 100 light switches, numbered by the positive integers 1 through 100.  There are also 100 children, numbered by the positive integers 1 through 100.  Initially, the switches are all off.  Each child k enters the room and changes the position of every light switch n such that n is a multiple of k.  That is, child 1 changes all the switches, child 2 changes switches 2, 4, 6, 8, …, child 3 changes switches 3, 6, 9, 12, …, etc., and child 100 changes only light switch 100.  When all the children have gone through the room, how many of the light switches are on?

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• Each of two players picks a different sequence of two coin tosses.  That is, each player gets to pick among HH, HT, TH, and TT.  Then, a coin is flipped repeatedly and the first player to see his sequence appear wins.  For example, if one player picks HH, the other picks TT, and the coin produces a sequence that starts H, T, H, T, T, then the player who picked TT wins.  The coin is biased, with H having a 2/3 probability and T having a 1/3 probability.  If you played this game, would you want to pick your sequence first or second?

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• 100 coins are to be distributed among some number of persons, referred to by the labels A, B, C, D, ….  The distribution works as follows.  The person with the alphabetically highest label (for example, among 5 people, E) is called the chief.  The chief gets to propose a distribution of the coins among the persons (for example, chief E may propose that everyone get 20 coins, or he may propose that he get 100 coins and the others get 0 coins).  Everyone (including the chief) gets to vote yes/no on the proposed distribution.  If the majority vote is yes, then that’s the final distribution.  If there’s a tie (which there could be if the number of persons is even), then the chief gets to break the tie.  If the majority vote is no, then the chief gets 0 coins and has to leave the game, the person with the alphabetically next-highest name becomes the new chief, and the process to distribute the 100 coins is repeated among the persons that remain.  Suppose there are 5 persons and that every person wants to maximize the number of coins that are distributed to them.  Then, what distribution should chief E propose?

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