Friday, May 8, 2015

The top 5 finalist essays, FQXi essay contest 2015

Here are the top 5 essays from the 40 finalists of this year's FQXi essay contest, based on the community ratings.

Unofficially, since FQXi didn't announce yet which of the more than 200 essays are the 40 finalists, although the announcement was expected since April 22. My essay is on the fourth place.

The finalists will be judged by a jury, who will decide the awards until June 6, 2015.

Monday, March 16, 2015

The Monty Hall problem, retold

The Monty Hall problem

The Monty Hall problem is inspired by an American television game show. There are three doors, and behind one of them, the host of the show, Monty, hides a car. Each of the other two doors hides a goat.

The contestant is asked to pick a door, so that if she finds the car, she wins the game (and the car). Since there are three doors, chances are $1/3$ that she picked the door behind which is the car. But Monty doesn't open yet the door, but he opens one of the remaining doors, revealing a goat. He then asks the contestant either to keep her original choice, or to switch to the other unopened door. The problem is, what should the contestant do?

The first instinct of anybody may be to think that since there are only two remaining doors, it doesn't matter if you switch the door or not, because the chances are $1/2$ in both ways. However, Marilyn vos Savant explained that if the contestant switches the doors, the chances are $2/3$. while if she doesn't switch them, the chances are $1/3$. This is counterintuitive, and the legend says that not even Paul Erdős understood it. You can find on Wikipedia some solutions of this puzzle.

An equivalent puzzle

I will present another, simpler puzzle, and show that it is equivalent to the Monty Hall problem.

Consider again three doors, one hiding a car. The contestant is asked to pick either one of the three doors, or two of them. What is the best choice?

Obviously, the contestant should better choose two doors, rather than one. Since if she thinks that the car is behind door number three, choosing also door number one will only double the chances to win.

But how is this related to the Monty Hall problem? Well, it is, because if you play the Monty Hall problem, you can pick two doors, but then tell Monty, you just tell you picked the remaining one. When Monty asks if you want to switch, then you switch to the other two doors, and since one is already open, you choose the remaining one. This means that choosing a door and switching is equivalent to choosing the other two doors.

So the Monty Hall problem is actually equivalent to having to choose one or two doors. Not switching is equivalent to choosing one door, and switching is equivalent to choosing two doors. So switching gives indeed probability $2/3$.

Saturday, March 14, 2015

Round squares exist

Bertrand Russell said that there are no round squares. But there are. Here are two solutions.

A circle-square

This is a square that is circle:

To make it, first make a paper circle and  a paper square, with equal perimeters:

Fold them a bit:

Then paste their edges together:

The common boundary forms a square that is circle. It is a square, because in the blue surface it has right angles and equal straight edges. It is a circle, because in the red surface its points are at equal distance from a point. In fact, its points are at equal distance from the center even in space, because the red surface is ruled, and all the lines pass through the same point. So the common boundary is also a line on the surface of a sphere.

Round squares in non-Euclidean geometry

Consider for example the geometry on a sphere. On a sphere, polygons are made of the straightest lines on the sphere, which are arcs of the big circles. So, there are squares on a sphere

Image from Wikipedia
This is a square, since its edges are the shortest and straightest lines on the sphere, they have equal lengths, and its angles are all equal. If one gradually increases the size of the square, the angles increase too. At some point, the angles become $180^\circ$, and the edges become aligned, forming one single big circle:

Image from Wikipedia

So, is it a circle? Is a square? Is a circle and a a square!