The 100 Prisoners And The Light Bulb

One hundred prisoners have been newly ushered into prison. The warden tells them that starting tomorrow, each of them will be placed in an isolated cell, unable to communicate with each other. Each day, the warden will choose one of the prisoners at random, and place him in a central interrogation room for an hour. A prisoner can be chosen any number of times; thus it could happen a prisoner is chosen multiple times while another is yet to be picked. The interrogation room contains nothing but a light bulb and a light switch. The prisoner can see whether the light is on or off, and – if he wishes – can toggle the light switch. He also has the option of announcing that he believes all prisoners have visited the interrogation room at some point in time.

If this announcement is true, then all prisoners are set free, but if it is false, all prisoners are executed. So obviously the announcement should only be made if the prisoner is 100% certain that it is true!

The warden leaves, and the prisoners huddle together to discuss their fate. This is the last time they can speak with each other! Can they devise a system that will guarantee their freedom?  

The Monty Hall Problem

Suppose you're on a game show, and you are given the choice of three doors: behind one door is a car; behind the other two are goats. Naturally your goal is to win the car, which is equally likely to be behind each door.

You pick a door (e.g. #1), and the host (who knows what's behind each door) opens another (e.g. #3) which has a goat. This means there are two doors left, one with a car and one with a goat. You are now given the opportunity to switch your choice (from #1 to #2). Is it to your advantage to do so?

Since you can't know which of the two remaining doors has the car, and since your initial pick had a chance of one-third, you might think that it does not matter. The chance is still ¹/₃ and you might as well stay with your original choice, right? Wrong! Switching actually doubles your chances to ²/₃.

The Triangle Dissection Fallacy

Take a look at the above picture, and agree with me that this is weird. We see two triangles, and based on the grid they are both the same size. A quick count of the grid boxes tells us that the surface of both triangles should be 13 × 5 ÷ 2. All the pieces are equal in size, yet by shuffling them around we suddenly have a spare grid box!

This does not make any sense. The act of rearranging the triangle pieces should not change the surface area. What's going on here?

The Unexpected Hanging

“A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the ‘surprise hanging’ can't be on a Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on a Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.”   ^source

An interesting paradox! There are many variations to this story. For example, another popular version is about a surprise examination in class. Resolutions from varying perspectives have been suggested, but the different schools of thought have yet to agree on the correct one.

The Birthday Paradox

Given a group of randomly chosen people, how big is the chance that two persons in this group share the same birthday? You can assume that each day of the year is equally probable for a birthday.

The answer might surprise you. In case of a group size of 23 people, the chance that there are two people in there with the same birthday, is already 50/50. A probability of 99% is reached with a group of only 57 people!

The String Around The World

Imagine I had a very long piece of string: long enough to wrap it around the equator of the Earth. And I'll do just that. Yikes — that's about 40,000 kilometers (or 25,000 miles) of string! I will make sure its pulled completely tight and connect both ends to each other: the result is that it lies flat onto the surface.

Now let me extend the string with just one measly meter. Compared to its total length that's not a lot, is it? Once again I pull it tight, but now, with the added meter, it has come off the ground just a tiny bit. Assume that this extra distance (between the string and the Earth's surface) is equally divided and thus the same all around the globe. How much would you guess this distance is? Surely this must be in the order of nanometers or whatnot, right?

Well, no. Turns out, the entire string has now come almost 16 centimeters off the ground.