You are on a game show and presented with 3 doors. Behind one is a car and behind the other 2 are goats. You want to choose the door with a car behind it, as if you do so, you win the car. You choose one door. Then, the host opens one of the other doors, which reveals a goat behind it. The host gives you a choice to either switch your door to the other one that’s still closed or keep your original choice. Should you switch doors? Answer: If your strategy is to stick to your original choice, your probability of choosing the door with the…


Brainteaser: 100 Prisoners in a Line and Extension
There are 100 prisoners. An executioner tells them that tomorrow morning, he will line them up so that each of them is facing the back of the head of another prisoner, except for one prisoner at the end of the line. In other words, prisoner 1 sees the back of the head of prisoner 2 as well as the backs of the heads of prisoners 3100, prisoner 2 sees the back of the heads of prisoners 3100, …, prisoner 99 only sees the back of the head of prisoner 100, and prisoner 100 doesn’t see any prisoners in front of him. The executioner tells them that he will put either…

Brainteaser: Blue Foreheads
100 people are in a room. All 100 of them are perfect logicians. They are told that at least one person in the room has blue paint on their forehead. They are told that once you deduce that you have blue paint on your forehead, the next time that the lights are turned off, leave the room. All 100 people have actually had their foreheads painted blue (but of course, each of them don’t know this at this point – they can only see the other people’s foreheads). The light is turned off, then on, then off, on, etc. What happens? Answer: So each person sees…

Brainteaser: Forehead Numbers
There are 3 people placed in a room. They all have perfect logic. The 3 people are told by a host that a number has been written on each of their foreheads. Each of the 3 numbers are unique, they are all positive, and they relate to each other such that A + B = C (i.e. one is the sum of the other two). In the room, each person can only see the other two people’s numbers, as they cannot see their own foreheads. Suppose you are one of the 3 and you see one person with “20” on their forehead and the other person with “30.” The…

The Theory of Interstellar Trade
The Theory of Interstellar Trade, by Paul Krugman (1978) Archived Assume we have two planets, Earth and Trantor, separated by a large distance, the traversal of which necessitates travel at velocities comparable to the speed of light. Assume that Earth and Trantor are in the same inertial reference frame, i.e. they are not accelerating with respect to each other. Assume that a spaceship traveling between the two planets travels at a constant \(v\). Let’s say that from the perspective of an observer on one of the planets, the time it takes for a spaceship to make the trip is \(n\). Then, the time it takes for a spaceship to…