Problems 181-200

Last update: Nov. 26, 2008

181. You are trapped in a small phone booth shaped room. In the middle of each side of the room there is a hole. In each hole there is a push button that can be in either an off or on setting. You can't see in the holes but you can reach your hands in them and push the buttons. You can't tell by feel whether they are in the on or off position. You may stick your hands in any two holes at the same time and push neither, either, or both of the buttons as you please. Nothing will happen until you remove both hands from the holes. You succeed if you get all the buttons into the same position, after which time you will immediately be released from the room. Unless you escape, after removing your hands the room will spin around, disorienting you so you can't tell which side is which. How can you escape? The fewest possible turns that I know if is seven. (solution).
182. There is a one acre field in the shape of a right triangle, with sides of length x and y. At the midpoint of each side there is a post. Tethered to the posts on each side is a sheep. Thethered to the post on the hypotenuse is a dog. Each animal has a rope just long enough to reach the two adjacent vertices of the triangle. How much area outside of the field do the sheep have to themselves? (answer). (solution).
183. Each day a man meets his wife at the train station after work, and then she drives him home. She always arrives exactly on time to pick him up. One day he catches an earlier train and arrives at the station an hour early. He immediately begins walking home along the same route the wife drives. Eventually his wife sees him on her way to the station and drives him the rest of the way home. When they arrive home the man notices that they arrived 20 minutes earlier than usual. How much time did the man spend walking? (answer). (solution).
184. Ten people land on a deserted island. There they find lots of coconuts and a monkey. During their first day they gather coconuts and put them all in a community pile. After working all day they decide to sleep and divide them into two equal piles the next morning. That night one castaway wakes up hungry and decides to take his share early. After dividing up the coconuts he finds he is one coconut short of ten equal piles. He also notices the monkey holding one more coconut. So he tries to take the monkey's coconut to have a total evenly divisible by 10. However when he tries to take it the monkey conks him on the head with it and kills him. Later another castaway wakes up hungry and decides to take his share early. On the way to the coconuts he finds the body of the first castaway, which pleases him because he will now be entitled to 1/9 of the total pile. After dividing them up into nine piles he is again one coconut short and tries to take the monkey's coconut. Again, the monkey conks the man on the head and kills him. One by one each of the remaining castaways goes through the same process, until the 10th person to wake up gets the entire pile for himself. What is the smallest number of possible coconuts in the pile, not counting the monkeys? (answer). (solution).
185. A 10 inch stick is thrown into a buzz saw and cut in two pieces at a random point. Each resulting piece is thrown into the buzz saw again, and each is again cut at a random point. What is the probability that all four remaining pieces are one inch long or greater? (answer), (solution).
186. On a game show there are three doors. Behind one door is a new car and behind the other two are goats. Every time the game is played the contestant first picks a door. Then the host will open one of the other two doors and always reveals a goat. Then the host gives the player the option to switch to the other unopened door. Should the player switch? (answer), (solution).
187. A column of soldiers one mile long is marching forward at a constant rate. The soldier at the front of the column has to deliver a message to the soldier at the rear. He breaks rank and begins marching toward the rear at a constant rate while the column continues forward. The soldier reaches the rear, delivers the message and immediately turns to march forward at a constant rate. When he reaches the front of the column and drops back in rank, the column has moved one mile. Question- How far did the soldier delivering the message march? (answer) (solution).

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(answer) (solution).

189.
The radius of the circle is 1. The triangle is equilateral. Find the area of each colored region.

(answer), (solution).
190.
There is a table in a dark room with 100 coins. You can't see anything nor feel which sides are up. 80 are heads up and 20 are tails up. Another table in the room has nothing on it. How can you get the same number of coins tails up on both tables with at least one penny on each table?

(answer)
191.
In double-zero roulette, what is the probability that any number will not have hit by the 200th spin?
(answer), (solution).
192.
In a game, players A and B must each secretly pick Rock or Paper. The matrix below shows how much player B must pay player A, according to each combination of choices. Player A wishes to maximize his expected win, and player B wishes to minimize his expected loss. What is the optimal strategy for each player?

Problem 192

Player A Player B

Rock Paper

Rock 7 3

Paper 2 11

(answer), (solution)
193.
In a game, players A and B must each secretly pick Rock, Paper, or Scissors. The matrix below shows how much player B must pay player A, according to each combination of choices. Player A wishes to maximize his expected win, and player B wishes to minimize his expected loss. What is the optimal strategy for each player?

Problem 193

Player A Player B

Rock Paper Scissors

Rock 6 0 6

Paper 8 -2 0

Scissors 4 6 5

(answer), (solution)
194.
What is the average of the lesser of two random numbers from 0 to 1?

(answer), (solution)
195.
What is the average of the least of n random numbers from 0 to 1?

(answer), (solution)
196.
You are playing a game with three people, you, an opponent, and a referee. Each will pick a real number between 0 and 1. The referee will chose randomly, with each pick equally likely. The player who comes closer, without going over, will win. If both go over, or both pick the same number, the game will result in a tie. You know your opponent will chose a number randomly between 0 and 1, each equally likely. What strategy should you employ?

(answer), (solution)
197.
You are playing a game with three people, you, an opponent, and a referee. Each will pick a real number between 0 and 1. The referee will chose randomly, with each pick equally likely. The player who comes closer, without going over, will win. If both go over, or both pick the same number, the game will result in a tie. What strategy should you employ?

(answer), (solution)
198.
You are playing a game with two perfect logicians. All three must secretly write down a positive integer. The one with the lowest unique integer will win $3. If all three have the same number, each will win $1. The logicians are selfish, and each wishes to maximize his own winnings. Communication is not allowed. What number should you choose?

(answer), (solution)
199.
Same question as 198, except in the event of a 3-way tie, nobody wins anything.

(answer).
200.
In Major League Baseball, the American League has three divisions, the East and Central divisions with 5 teams each, and the West division with 4 teams. Four teams in the National League will make the playoffs, the three division leaders, and a wild card team. The wild card team is the team with the best record in the league, not including the three division leaders. Assuming all teams are equally good, what is the probability of a team in each division making the playoffs? Assume any ties are broken randomly.

For extra credit, also do the National League, in which the East and West divisions have 5 teams, and the central division has 6 teams.

(answer), (solution)

Michael Shackleford, ASA

Problem 193
Player A	Player B
Player A	Rock	Paper	Scissors
Rock	6	0	6
Paper	8	-2	0
Scissors	4	6	5