The Monte Hall 'Make a Deal' Problem

One of the most controversial recreational problems is the Monte Hall Problem. The idea stems from the original game show entitle 'Let's Make A Deal.' The problem appeared first in an article by Steve Selvin in the journal American Statistician.

The problem gained attention when it became the object of a Marilyn Vos Savant column in Parade magazine where it was stated in the following form:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Vos Savant says that the probability of obtaining the car improves if you switch. Many people find this answer runs counter to intuition. This answer like all answers depends on the assumptions that are made especially in regard to the statement

the host, who knows what is behind the doors,...

If we assume that the strategy of the host is to minimize the number of cars given away, then he will not always show the contestant a goat, but the common assumption in the problem is that the host will always show a door that has a goat behind it.

Let us assume that the host does show a goat to the contestant no matter which door is originally chosen. The probability that the car was originally chosen is 1/3rd. The probability that the contestant did not chose the car is 2/3rds, so it follows that the contestant will obtain the car 2/3rds of the time by switching after the host shows a goat.

It is easier to see this relation by considering the question when the number of doors is large. For instance, suppose there are 100 doors. We further assume you pick one door at random and then the host opens 98 doors with goats in them. Obviously, the probability is 1/100 that you originally picked the right door and the probability that the car is behind the remaining door is 99/100.

But suppose the host wishes to minimize the likelihood that the contestant will get the car. What strategy ought he to pursue ?

The Lottery Paradox

Consider the following lottery problem:

• one million tickets are sold for one dollar each
• one ticket is chosen and the owner of the ticket wins the entire pot

The probability of any one ticket winning is 1/1000000. In other words, the probability of winning is nearly zero but the collective sum of these probabilities is 1.

The paradox involves the rationality of belief. It is rational to believe that one individual ticket may win but the conglomeration of this rational belief leads to the belief that no ticket can win which is clearly a false belief.

The paradox is related to the question of knowledge and the the accumulation of nearly perfect knowledge which leads to a false conclusion. This is not a mathematical paradox in the sense that nearly perfect knowledge of the truth of a set of hypothesis does not even allow one to conclude that even one of the hypotheses hold.

On the one hand, we have mathematical methods which give one result, but on the other hand, we have a philosophical approach which leads to confidence, not certainty, in another result.

A practical example of this problem occurs in digital communication. In a communication channel, like the channels used to distribute web-pages over the internet, there is a small possibility that a digit can be changed during transmission. While the probability is extremely small for any one digit, the conglomeration of millions of digits implies that eventually an error will occur. Internet protocols generate a method for correcting the errors, when they occur, in most cases.

The Sleeping Beauty Paradox or the Computer Decision Tree Problem

The Sleeping Beauty Paradox grows out of the Absent-Minded Driver Paradox which in turn stems from a computer programming problem studied by M. Piccione and A. Rubenstein in the journal Games and Economic Behavior, 1997. We first consider the form of the problem ( rather than paradox) as stated in the so-called Sleeping Beauty Paradox. Here is the problem as originally posed by Jamie Dreier in an internet recreational puzzle discussion group:

• We plan to put Beauty to sleep by chemical means, and then we'll flip a fair coin, i.e. the probability of heads is 1/2 and the probability of tails is 1/2.
• If the coin lands heads, we will awaken Beauty on Monday afternoon and interview her. If it lands tails, we will awaken her Monday afternoon, interview her, put her back to sleep, and then awaken her again on Tuesday afternoon and interview her again.
• In each case the interview is to consist of the one question: what is your credence now for the proposition that the flipped coin landed heads?
• Under no circumstances will Beauty be able to tell which day it is, nor will she remember whether she has been awakened before. She only knows the details of the experiment.
• What credence should she state in answer to the interview question?

In the form of the Absent-Minded Driver one version allows that there consequences for poor choices. Hence, the problem involves what is called the maximization of the expected outcome. As stated, the Sleeping Beauty Problem asks the question: what credence should Sleeping Beauty give to the two outcomes of the coin toss?

This question does not make clear what is meant by credence. It appears that rather than restating the Absent-Minded Drivers Problem, the Sleeping Beauty problem is inspired by it and only closely related to it. Most interpretations of the Sleeping Beauty Problem give a solution in terms of probability theory and this seems to be a reasonable approach. Some interpretations appear to consider the question of credence to be related to expected correct answers.

How should the game be interpreted and what is the correct answer under the various interpretations?

More on the Unexpected Hanging

See the problem "The Inevitable Surprise" below for the initial discussion.

Some authors consider the problem of the unexpected hanging to be an example of an actual paradox, meaning that the sentence of the judge is self contradictory. Assuming that the judge has seven days to choose from, we have shown using game theory that the the sentence is not self contradictory. But consider the reduced problem where the judge pronounces the sentence on Thursday saying thatt the hangman will appear at the jail cell on Friday or Saturday and the hangman will be unexpected.

The assumption are:

1. The judge's statement is true;
2. Predicate logic holds.

Under these assumptions we can see that

• If the judge chooses Saturday, then the hangman will be expected;
• The judge must choose Friday, so by the previous statement the prisoner ought to expect the hangmen on Friday.
• If the hangman can be expected on Friday, then the judges statement is false and so the statement is self-contradictory.

Some authors claim that the reason for this paradox is that the judge's sentence is self-referential. More on this at the Mathematics Resolution link at the top of the blog.

The Inevitable Surprise

In Martin Gardner's book The Unexpected Hanging and other Mathematical Diversions, a logic puzzle is proposed according to which a judge pronounces on a Saturday a sentence of hanging. In addition, the sentence will be imposed in such a way that the hangman will appear at the jail cell door at noon on some day during the following week beginning with Sunday and extending to Saturday. Finally, the appearance of the hangman will occur on a day that the prisoner does not expect.

There are many versions of this puzzle including the original version in which an air raid warden announces that there will be a surprise drill at 8 p.m. some night in the next seven days. Of course, such a statement could be said of a pop quiz or any military drill. Part of the controversy in each case is the question as to whether this sort of statement can be absolutely true.

Gardner does a very good job of outlining the paradoxical nature of the problem. On the one hand, the prisoner's lawyer argues that if the prisoner is not hanged by Friday, then the predicted surprise can not occur on Saturday at noon since then the judge's statement would have been false. Hence the purported contradiction to the judge's pronouncement of sentence. But we must assume that the judge's sentence is a statement of fact and so the prisoner must be hanged on some day before Saturday. If this must be true, then we can conclude that if the prisoner is not hanged by Thursday, then he will know that the judge can not delay the hanging until Saturday and so will know for a certainty that he can not be hanged on Friday.

Likewise, if the prisoner is not hanged on Wednesday, he will know that he cannot be hanged on Friday or Saturday and so he must expect the hangman to come on Thursday. But the hangman can not come when expected, so He can not be hanged on Thursday, Friday or Saturday.

The argument continues in this fashion so that it is clear to the prisoner that there is no day which the judge may pick for which the prisoner will not expect the hangmen. Secure in this knowledge the prisoner does not expect the hangman when he appears on Wednesday at noon. See comments for a discussion

Invisible objects

There are a lot of unscience fiction stories about invisibility cloaks and so forth, bu is it possible to build an object that is for all intents and purposes invisible.

Mathematicians say it is so and we give here a very simple example of an invisible object. Our definition of invisible is different from transparent. We would say that light passes through a transparent object. We will say that light is diverted around an invisible object.

In this case we will consider light coming in from infinity so that all the light rays are parallel.




Now observe that light from infinity is diverted around the object if we have constructed the angles properly. This construction involves isosceles triangles with base angles of 30 degrees such that the bases of the two facing triangles are a distance of 1 apart and the length of each base (the vertical line) is sqrt(3)/2.

Under these circumstances, can you show that the path of the line at the beginning and end are in the same direction (or line).

See Mathematics Resolution on invisible objects for more information.

A Simple Railroad Car Problem

Consider the simple railroad car problem

Two trains headed by engines wish to pass each other so that the trains are in the same order
after reassembly. There is one siding which will hold one car at a time. The engines may disconnect and go into the siding or they may push a car into the siding but only from the right side of the diagram.

This is similar to a sliding block puzzle except that that certain blocks or cars can only move if attached to the engines.

Give a solution to the problem. Is your solution the best (shortest number of moves) possible where a move is a position where a car has been put into the siding or removed from the siding.

Can you complete this task?

A supertask is a countably infinite sequence of tasks to be performed in order, say

{T₁,T₂,T₃,...}

such that we first perform task T₁ and then task T₂ and so forth.

As an example one might consider the case of Zeno's Paradox where an arrow is shot at some target. The arrows first task is to complete the journey half way to the target. The second task is to complete the remaining journey half way to the target and so on.

Now consider the case of Jon Perez Laraudogoitia's beautiful supertask: There are a sequence of unit point mass balls stationary at x=1/2,x=1/4,x=1/8 and so on. A unit point mass at x=1 and time t=0 is moving with a velocity of 1 unit per second to the left (i.e. toward x=1/2).

We assume that there is an elastic collision at x=1/2 and so the moving ball is left stationary at x=1/2 and the ball at x=1/2 moves with velocity 1 unit per second in the direction of the ball at x=1/4. It is clear that at t=1 all the balls have been hit by the preceding ball.

Assuming that all the collisions are elastic and behave in exactly the same manner as the first collision, is there a ball that passes through x=0?

The Impossible Problem of Hans Freudenthal

Hans (Freudenthal) concocts a little game for his friends Sam and Patty. Choosing two numbers each greater than one but also less than 30, he secretly tells Sam the sum and he secretly tells Patty the product. Sam and Patty are not allowed to discuss the values that they are given and yet they are challenged to find the two original numbers.

You witness the following exchange:

Patty says, I can not find the numbers.

Sam then says, I knew that you could not find them.

Patty responds, Oh, then I know what the numbers must be.

Sam replies, Now I too know the numbers.
How is this possible? Of course, Patty and Sam have the advantage of knowing more than we do. Can you solve this problem with the little information given?

Start with the hint that the sum is less than 30.

Taxi-Cab City: A Travelogue

I have been told that there is a city hidden in a valley of the Andes mountains by the name of Catcaxitybi (by their language cat-ca-chee-tih-bee) that is laid out in a perfect square. And though most modern conveniences are absent they have managed to port a good number of cars over the intervening mountains. But because cars are still quite scarce it is decreed that those cars that run shall be taxi-cabs. Perhaps for this reason, but possibly not, they have developed a subtle form of geometry. Pythagoras and his right triangles intrude into their calculations not at all. Of Euclid’s cowl they have never heard, though there is in that city a popular headdress that appears to be made of cloth in the approximate shape of the cowl called a cloules cwido pronounced clue-less-kwid-doe.

Euclid’s cowl gives a demonstration of Pythagoras’ theorem by showing that the colored triangles all have the same area.

Oddly enough the word for stranger in the native tongue is cwido.

Indeed, the hypotenuse is such a strange notion to them that there is a legend told about the town wherein it is said that a scientific team set out in search of this hidden city to correct this most grievous fault but turned up missing (as the locals say). By this they mean that the geometers were lost in the jungle. A search party was formed in ancient native costume and sent to mislead the strangers back to civilization and its geometry.

The city is 25 blocks by 25 blocks. The streets running north and south are numbered First Avenue to Twenty-sixth Avenue going from west to east. The streets running east and west are called A Street to Z Street going from south to north.

Two different equilateral triangles in taxi-cab geometry

Communication in the city is carried out by the use of homing pigeons called Leoc saw shireft pronounced Lok-saw-sure-eft. These birds apparently do not care for the ways of crows and their methods of flying but strictly follow the street routes to their proper destinations.


The buildings are tall and all of regulation height, width and length sporting luscious tropical gardens atop each one. There are strange gardens with overhanging vines, or curiously those that look like the grounds of an English Manor house. Often there are beautiful gardens full of orchids but then there are also many vegetable gardens providing produce for the town folk. Some buildings are covered with apple orchards altogether giving the city a distinctive look from above or rather a look indistinguishable from the surrounding jungle.

A view from above

The Catcaxitybians play a curious game called chorecks reshecs (chow-rakes-ray-shiks) played on a board of sixty four squares. There are eight white pieces usually of the same fashion and eight black pieces facing each other from opposite ends of the board. A chess board and pawns or checkers can be used to play the game. A piece may move a distance one in a single direction or a distance 2 if it changes direction. A piece must be taken if it is on an adjacent block to any opposite piece. The object of the game is to be rid of your pieces first. The city boasts of many world champions in the game even though it is the only place of my acquaintance that suffers to play such an unorthodox game wherein the loser is forced to win. It is a tradition to recite the maximum distance between your pieces and your opponents pieces before each move with a time a penalty for incorrect recitations. In friendly games with foreigners this custom is usually abandoned. When the last piece is taken the winner says: ihndet belf. The word is in an ancient tongue which no one here seems to know.

White to play and win.

Each city block is exactly one tenth of a melotikre (mel-low-teek-rah). The fares are fixed at a price of xiced frepi ( chee-sid freh-pee) per block in the local currency. A frep is about one quarter of a dollar at current exchange rates, though the exchange rates are determined by the locals and probably have nothing to do with actual values of the currency. I asked my native friend for one good reason that this should be so. He replied that the locals justify this strange practice by pointing out the curious fact that international exchange rates have nothing to do with the relative values of the currencies involved. I told him that it was a reason but not a good one. There are no meters in the taxi-cabs as all the drivers are good at mental arithmetic and have an innate sense of geometry, such as it is in Catcaxitybi. Thus if you get in a taxi-cab at Fifth and D and wish to go to Thirteenth and R the driver will immediately quote the price of 22 xiced frepi.


I have listed below the entrance locations for several buildings in the city. If you can compute the fares mentally, you could become a taxi-cab driver in the city. I am afraid I failed the entrance exam in miserable fashion for I am too slow and must use my fingers to count.

Shroe Eapou - Tenth and G;


Ranem Trod - Twelveth and Q;


Dacherlat - Twentieth and B;


Mafer Karstrem - Thirteenth and N

Perhaps you noticed that there are several anagram phrases in the description of the taxi-cab city above. The anagrams are associated to the words in the native language.

Can you decipher each anagram?

Can you do taxicab geometry in your head?

Can you find another kind of equilateral triangle with side length 6?

What does a square look like? What does a circle look like?

Is it possible to fit a square peg in a circular hole in this geometry?