A Prisoners Riddle

I just heard a nice riddle and wanted to share.

Prisoner A and prisoner B are given an integer between 1 and 100. Each prisoner only knows his own number, but also that the two numbers are consecutive. For example, if prisoner A got the number 60, he knows that prisoner B got either 59 or 61. At the end of every hour each prisoner can choose to guess the number of the other. If either prisoner guesses correctly, they both go free. However, if either prisoner guesses wrong, they both get executed (even if at the same time the other guessed correctly). Both prisoners can choose not to guess for as many hours as they like.

The two prisoners are in different cells and cannot communicate in any way. Also, they did not have time to coordinate a strategy in advance. Each can only assume that the other prisoner is intelligent. Can they always guess correctly? After how many hours?

I think it would be nice not to post solutions in the comments. Although you can just write the number of hours you got.

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A Boxes Riddle

I just recalled a nice mathematical riddle. I can’t remember where I originally read it, but it was likely in one of the blogs that are in the blogroll to the right.

Riddle. A game takes place where person A and person B are on the same team while person Z is their adversary. There are 100 boxes and 100 notes containing the numbers 1 to 100 (that is, each number is on exactly one note). The game goes as follows:

• First, only Z and A are in the room. Z places one note in each box.
• A sees the actions of Z and may afterwards pick two boxes and switch the notes in them (with each other). He may only perform one such switch.
• Z sees the actions of A and then chooses a number N between 1 and 100.
• A leaves the room while B enters it (they cannot exchange information during this process). Z tells B the number N.
• Finally, B needs to find the box that contains the note with the number N. For this purpose, B may open up to 50 boxes.

If A chooses the 50 boxes at random, his probability of success is obviously 0.5. Do A and B have a strategy for increasing their chances? Or does Z have a strategy for which 0.5 is the best possible?

Feel free to write the answer as a comment. However, I think that it would be nice not to provide a full explanation here (that is, only to write whether it’s 0.5 or what better probability you can get).

(Later edit: I seem to be getting senile! I just noticed that I wrote the same riddle in a post two years ago…)

Quoting Rota

“What can you prove with exterior algebra that you cannot prove without it”? Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, idèles and Grothendieck’s schemes, to mention only a few. A proper retort might be: “You are right. There is nothing in yesterday’s mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.”

This paragraph is taken from Indiscrete Thoughts by Gian-Carlo Rota, in a section discussing how Grassmann‘s work was ignored for decades. And yes, I am quoting other people because I have no time for more serious posts. Hopefully these will return in about six weeks. In the meantime, another Rota from the same book:

One day, in my first year as an assistant professor at MIT, while walking down one of the long corridors, I met professor Z, a respected senior mathematician with a solid international reputation. He stared at me and shouted “Admit it! All lattice theory is trivial!”

Being a group theorist can be dangerous!

The Maoist movement decided that group theory was a reactionary subject of the old regime, and started protesting at the increasing number of professors in the subject being appointed. Demonstrations erupted outside of the maths department with protesters holding placards demanding ‘No more group theory’. One new appointee in group theory was frightened off and took a job elsewhere. During one demonstration, the students scaled the outside of the building and scrawled `Group Theory Department’ on the wall.

This paragraph is taken from the non-fiction book Finding Moonshine by Marcus du Sautoy. These events took place in Germany in the early 70’s.

A Noah’s Ark Joke

The Flood has receded and the ark is safely aground atop Mount Ararat; Noah tells all the animals to go forth and multiply. Soon the land is teeming with every kind of living creature in abundance, except for snakes. Noah wonders why. One morning two miserable snakes knock on the door of the ark with a complaint. “You haven’t cut down any trees.” Noah is puzzled, but does as they wish. Within a month, you can’t walk a step without treading on baby snakes. With difficulty, he tracks down the two parents. “What was all that with the trees?” “Ah,” says one of the snakes, “you didn’t notice which species we are.” Noah still looks blank. “We’re adders, and we can only multiply using logs.”

                                                                        Letters to a young mathematician / Ian Stewart.

Books and quotes

This post is about two popular-mathematics books the I recently read:

MY BRAIN IS OPEN: The Mathematical Journeys of Paul Erdős / Bruce Schechter

On one occasion, Erdős met a mathematician and asked him where he was from. “Vancouver,” the mathematician replied. “Oh, then you must know my good friend Elliot Mendelson”, Erdős said. The reply was “I AM your good friend Elliot Mendelson.”

Several years ago I read another biography of Erdős by Paul Hoffman, which was nice, though focused mainly on Erdős’s eccentricities. I think that Schechter’s biography would appeal more to mathematicians, since it consists of a lower dose of eccentricities and of a higher dose of details about the mathematical world. For example, it was very interesting to read about how a Hungarian mathematics journal for high school students helped in the nurturing of many great mathematicians, and about the events that led to the elementary proof for the prime number theorem by Selberg and Erdős. I definitely recommend this book.

For those who are looking for an even lower dose of eccentricities and more details about the mathematical world, I recommend the piece “In and Out of Hungary, Paul Erdös, His Friends, and Times” by László Babai. It can be found in volume 2 of Combinatorics, Paul Erdös is eighty, and it contains many details about the Hungarian mathematical world (and about 20-th century Hungarian history in general).

Mathematical Cranks / Underwood Dudley

Represent heaven, the home of God, as a vector space of infinite dimension over some field known to god but unknown to us, in which the activities of God are quantifiable. Lengths will be measured (Revelation chapter 21 verse 15) in the usual way, as the square roots of the inner self-products of vectors (assuming heaven to be euclidean).

Was Disney trying to kill mathematicians during the 1930’s?

It’s time for a break from the technical posts, so the only equations in this post are in the following figure. Have you ever noticed that the deaths of two of the 20th century’s top mathematicians were related to Disney’s 1937 film “Snow White and the Seven Dwarfs”?

A more optimistic Disney-mathematics connection.

The more well-known story is the one of Alan Turing. It appears that Disney’s Snow White was Turing’s favorite film. Turing was especially fond of the scene where the wicked witch holds the poisoned apple, and used to chant the witch’s song

“Dip the apple in the brew / Let the Sleeping Death seep through.”

Eventually, Turing reenacted the following scene, committing suicide by taking a bite out of a poisoned apple (injected with cyanide).

Alan Turing, Kurt Gödel, and Mickey Mouse.