50 Prisoners and Hats

A king has fifty prisoners. He decides to play a game to give them a chance to win their freedom. The next day, he will line up the prisoners in a straight line, all facing the same direction. He will then place either a white hat or a black hat on each of their heads. There is no pattern to how he will place the hats and there is no known set number of each color hat. He could choose any number of black hats and any number of white hats, as long as each prisoner has a hat on his head. Once in a line, the prisoner at the back can see all the hats in front of him, and the prisoner in the front can not see any of the hats. The king will start at the back and allow the last prisoner to guess his hat color. He can only say "white" or "black", and he can not signal with timing, pitch, hand motions, etc. Only "white" or "black" in a monotone voice, or they will all be killed. If the prisoner guesses correctly, he will be set free. If not, he will be killed and the king will move forward to the next prisoner in line. He will continue this way all the way down until all the prisoners have attempted to guess their hat color. During the exercise, each prisoner can hear what happens to all the prisoners behind him (whether each lives or dies). The night before, the prisoners meet together and come up with a strategy where at MOST, one of them will die. What is this strategy that guarantees to save at least 49 of them?

Pirates and Treasure

Five pirates stumble across 1000 gold coins. They decide to divide up the coins amongst themselves in the following way. The first pirate will suggest a distribution of the coins. Let's say he suggests: 100 for pirate 5, 100 for pirate 4, 100 for pirate 3, 100 for pirate 2, and 600 for himself. After making a suggestion, all the pirates will vote on whether they agree to the distribution. If a CLEAR majority agree with the distribution, this distribution stays and the game is over. If not, then the pirates kill the first pirate, and the second pirate suggests a distribution for the remaining four pirates. Once again all the pirates vote and only if there is a clear majority does the second pirate live and the distribution stay. Note that a clear majority means over 50%, so if there are four pirates total then three half to agree for the pirate making the suggestion to live. This continues until a distribution is agreed on by a majority of the pirates. In addition to this set-up, all of the pirates have the same list of priorities. First, each is interested in self-preservation, meaning he will vote yes for a plan if he believes the alternative would result in him eventually dying. Second, the pirates are greedy, meaning that they will vote against a plan if they believe they can get more coins later on. Finally, all the pirates are bloodthirsty. This means that if they believe they can live and get the same number of coins later on, they would rather see another pirate die. Given these rules, what is the most number of coins that the first pirate can suggest for himself to guarantee that the distribution is accepted and he lives?

Duck in Pond

A duck starts off in the center of a perfectly circular pond with radius r. On the edge of the pond is a cat. The cat can run four times as fast as the duck can swim, but can not enter the water. Is it possible for the duck to reach any edge of the pond without the cat being there to catch it? If no, provide a proof as to why it can not be done. If yes, provide the strategy that the duck must use.

2 Ropes

You are given two ropes, each of which burn in one hour. However, the ropes do not burn linearly, so simply because one is half way through based on distance does not mean that thirty minutes have passed. The ropes also do not necessarily burn in the same way as each other. Using these two ropes, how can you measure exactly 45 minutes of time?

Two Coin Flippers

Two people are standing across from each other with a solid wall in between, so they can not see each other. Each has a fair coin. At each turn of the game, each will flip his coin and allow it to land on the floor. At the same time, each will try to guess out loud whether the other person's coin is heads or tails. What is a strategy that maximizes the probability that BOTH players are correct, and what is this probability?

Gameshow

You are on a gameshow. The host shows you three closed doors. Behind one of the doors is $1 million. Behind the other two doors is nothing. You begin the game by choosing one of the doors. It remains closed, but the host opens one of the other two doors and shows you that there is nothing behind it. In order to maximize your chances at winning the money, is it best for you to keep the door you originally chose, switch to the other closed door, or does it not make any difference?

10 Machines and 1 Weigh

You have ten machines that produce identical looking and feeling coins. Nine of the machines produce coins that weigh one ounce. However, one of the machines produces coins that weigh 0.9 ounces. You also have a scale with you (a digital scale, not the Libra kind). You can use the machines to produce any number of coins from each of them. Using only ONE weigh, how can you identify the machine with the lighter coins?

3 Prisoners and Hats

A king has three prisoners. He decides to give them a chance to earn their freedom. He has a total of three white hats and two black hats. The prisoners know that the king has these five hats total. The next day, the king says he will blindfold all three prisoners and place a colored hat on each of their heads. He will then remove the blindfold from the first prisoner and let him look around at the other two hats. If the prisoner can guess his hat color with 100% certainty, the king will let them all go free. If not, the king will put the blindfold back on the first prisoner and remove the blindfold from the second prisoner. The second prisoner now has a chance to see the other two hats and if he guesses his own hat color with 100% certainty, the king will let them all go free. If not, the king puts the blindfold back on the second person and allows the third person to guess his own hat color (without ever removing the blindfold). When the game is played, the first prisoner does not make a guess. The second prisoner, knowing the first did not guess, does not make a guess. When it gets to the third prisoner, who knows the first two did not guess, he is 100% sure what his hat color is and guesses it correctly, setting all the prisoners free. What is the third prisoner's hat color and how did he know it for sure?

Twelve Marbles

You have twelve marbles that look and feel identical. Eleven of the marbles weigh exactly one ounce. One of the marbles weighs a different amount, and you do not know it is lighter or heavier than the others. You also have a scale (the Libra kind, not the digital kind.) Using no more than THREE weighs, how can you guarantee to find the defective marble?

Fork in the Road

You are traveling through the woods and you come across a fork in the road. One of the paths leads to your destination, and the other path leads you into a trap. At the fork, you see two men. One always tells the truth, and the other always tells lies. However, you do not which man tells the truth and which one tells lies. What is ONE question you can ask that, regardless of which one answers, you will be able to know which path will take you to the destination?