To recap the puzzle:

A man walks into a casino and spots an interesting new game. He is presented with two buttons, either of which he can push. One of the buttons will win him a certain amount of money and the other will win him twice the amount, although he doesn’t know what the amount is. So he pushes one of the buttons at random and is surprised and very pleased to learn he just won $1 million. Suddenly, the machine presents him with two new buttons. He can push one to keep his original choice, keeping the $1 million. Or he can switch his choice to the button he didn’t push. What would you do and why?

When the man is shown how much he won when he pushed the first button randomly, that revealed some information: either the two rewards are $500K and $1 mil or they are $1 mil and $2 mil, but we don’t know which. Given that information, we can devise a strategy to determine whether we should switch our choice.

Suppose we draw a random number from anywhere from the real line using some probability distribution and we take that random number as a proxy for the amount behind the other button. For example, if we randomly selected $750K we would not switch, since $1 mil is bigger. But if we randomly selected $1.5 mil, we would switch, since $1.5 mil is larger than $1 mil. If we happen to select the random number and it falls in the interval of the true rewards, we will guess whether to switch correctly with certainty. And if we draw that number outside the interval, then we will guess correctly half the time.

Let’s formalize that. Suppose we have a pair of prizes (X, 2X) and let S be the event that we switch correctly using our strategy. We choose our random number from Z that comes from some distribution. Then,

But,

So,

which is greater than 50%.

We need to select our underlying probability distribution cleverly then. Suppose we choose a normal distribution with mean $1 mil and standard deviation $500K. That choice should guarantee good probability mass around the intervals (500K,1 mil) and (1 mil, 2 mil).

If it turns out that we are in the interval (500K, 1 mil), then the probability that Z is in (500K, 1 mil) is about 0.34 and the probability of choosing correctly is 1/2 + 1/2(0.34) = 67%. However, if the interval happened to be (1 mil, 2 mil), then the probability that Z is in (1 mil, 2 mil) is 0.48 and the probability of choosing correctly is 1/2 + 1/2(0.48)= 74%.

Suppose there are two trolls at this fork in the road. One always tells the truth and the other always lies but you don’t know which is which. You have only one gold coin to pay one of them to answer a question. What question will you ask one of them to be 100% sure of which fork to take?

Here’s a much harder variation: You are lost in a country in which one third of the inhabitants always lie. The other inhabitants tell the truth, but only three quarters of the time. Desperate to get out of the country, you come to a fork in the road guarded by a troll. One way leads out and the other to certain death, but you don’t know which is which. You can ask the troll which way is out if you pay him a gold coin. You have four gold coins.

1. You pay the troll a gold coin and he advises you to go right. Does that answer help you to decide?

2. You pay the troll another gold coin and he advises you to go right again. Does that answer help you to decide?

3. You pay the troll another gold coin and he advises you to go right yet again. Does that answer help you to decide?

4. You pay the troll your last gold coin and he advises you to go left this time. You must make a final decision. What should you do?

5. What is the chance you get out of this country alive after you’ve paid your last gold coin?

Solutions can be found here.

You and your friends want to play a game of craps, which requires two fair dice. The problem is that you have three dice, two of which are fair and one of which is loaded. You don’t know which of the dice are fair and which one is loaded and you don’t know how the loaded die has been loaded. Is there a way to roll the dice to have a fair game of craps? The solution is here.

Rose and Colin play a game where each have a large stack of coins, which they are allowed to arrange in advance with any sequence of H and T without letting the other player see the sequence. Once they have done this, they each simultaneously place a coin from the top of their stack onto the table. If HH shows up, Rose gives Colin 9 cents. If TT show up, she gives him 1 cent. However, if HT or TH shows up, then Colin gives Rose 5 cents. The game seems fair at face value. But is it fair? What do you think? How do you analyze this game? The solution is here.

There are N = 2ⁿ-1 people who are offered a challenge. If they win, they each get a million dollars. The men are blindfolded, and white or black hats chosen randomly are placed on their heads. The blindfolds are removed so they can each see the color of the hats on the other N-1. They must either guess the color of the hat on their heads or pass. If all pass, they lose. If one gets it wrong, they lose. If the answers are either pass or correct, with at least one correct, they win. What is the probability they will win a million dollars each? (Problem and Solution from Rutgers mathematics Professor Siddhartha Sahi, one of the original founders of our science club in Princeton and a colleague and friend of Gyan’s). The solution can be found here.

You may have seen examples of this problem on the internet. The usual way the problem is posed is the following:

You have 3000 bananas that you want to transport to a town 1000 kms away, using only a camel as the mode of transportation. The camel can carry a maximum of 1000 bananas at a time but eats one banana every km it travels while carrying bananas. If the camel is not carrying anything, he does not need to eat. What fraction of bananas can you transport?

The problem is not trivial because if you load up the camel with 1000 bananas each time, he will have eaten all of them by the time you get to the end of the journey.

We will solve this problem as it is usually posed and and then use it to find an exact solution to the general case where the camel can carry a load c, the distance to cover is d and the number of initial bananas is N = c x M. The questions we will ask are: How many bananas can be transported for finite N and what fraction of bananas can be transported in the limit when N tends to infinity?

The solution can be found here.

There is an old puzzle you may have heard of, which we will discuss a variation of. A group of mathematicians was sentenced to five years imprisonment for falsely claiming to have solved the Riemann Hypothesis. They are kept in solitary confinement most of the day, which is hardly cruel and unusual punishment given they have full access to the mathematics library and a whiteboard in every cell! But they would rather be able to discuss their findings with their colleagues over coffee.

A few days after their sentence begins, the warden, also a mathematician, called a meeting and made the following proposal: the mathematicians can decide to enter a contest in which the warden will select a prisoner randomly from solitary confinement every day and put him in a room. In the room is a single light switch, which will be in the off position the day the game begins. The selected prisoner can decide to turn the light on or off, but other than that, there is nothing to do in the room. The prisoner is not allowed to touch anything in the room or attempt to leave any kind of message. The other prisoners will not know who has been selected on any day and they will not be allowed to communicate in any way after the game begins. If one of the prisoners can correctly announce when all the mathematicians have been in the room at least once, the warden will free them all immediately. However, if they decide incorrectly, he will double their sentence to ten years. The warden gives the prisoners one hour to confer. Should the prisoners take the deal?

This is a two part puzzle: 1) Is there a way for the prisoners to infallibly decide whether all prisoners have been in the room at least once, even though they are not allowed to communicate? and 2) Should they take the deal even if they can figure out a solution to question 1? This puzzle as typically presented stops with question 1. Question 2 is more interesting. If it takes longer than five years to decide whether all prisoners have been in the room, it’s not worth it to take the deal.

See if you can solve these. The solution to the first question is here. The solution to the second puzzle is for more advanced students and can be found here.