April/May Question – Sneaky Dog

This is the last question for the Thinking Corner, and it is an interesting one indeed.  The first part requires careful thought, and the second part requires deep thought.  I hope that if you have gone through the due diligence this question demands, it will be as rewarding for you as it has been for me.  In particular, the second part of the question helped me understand not only the nature of information, but a lot about how society functions, and more importantly, a chance for self reflection.  Here is the question:

There is a village with 100 residents.  They all have extremely sharp minds, but also are very eccentric as they do not communicate with each other. Everyone has a dog, and everyone works at the sausage factory. An ongoing problem in the village is that some dogs sneak out and steal sausages when their owners are asleep. Through a bizarre arrangement, everyone has slightly different work and sleeping hours, so each resident knows exactly if everyone else’s dog is stealing, but not his own. The rule of the village is that if someone knows that his dog is stealing, he has to kick his dog out of the village the next morning; however, since nobody communicates with each other, this never happens.

One evening, the factory boss visits the village, gets everyone together and says, “There is sausage theft going on by at least one dog. That is all”. For a while, nothing happens. Sausages keep getting stolen. On a certain day after the announcement, 30 dogs get kicked out of the village.

This question has two components. The first is figuring out when and why the 30 dogs got kicked out, which is difficult enough in itself, but can be worked out with careful thought.

The second question is far more difficult. Remember that everyone already knows that dogs are stealing sausages. What the factory boss said, in fact, did not seem to impart any new information. Yet as we all know, if there is insufficient information to solve a problem, it will remain unsolvable until new information is given. Some additional information must have been given by the factory boss. What exactly is that new information?

The answer to the second question is subtle, requires deep thought and reflection, and surprisingly, reveals a lot about ourselves and human nature.

Answer to Feb/Mar Question – Unexpected Hanging

This is a rare example of a paradox that is also humorous. Although it does not initially seem worthy of serious discussion, surprisingly enough no fewer than 200 papers have been published on this paradox. Naturally, many of them start by dismissing other views and claiming that theirs is the long-awaited solution, the nail in the coffin.

What exactly, is wrong with the prisoner’s reasoning? There are two main approaches to resolve the paradox, logical and epistemological (how we acquire knowledge). The logical approach breaks down the reasoning by examining the basis (axiom) used for the argument to:

The prisoner will be hanged next week and its date will not be deducible in advance by using this announcement as an axiom.

Which is a self-referential statement and cannot be used to construct a valid argument. OK, the explanation is valid, but BORRRR-ING. The other approach is much more interesting:

The epistemological approach focuses on the meaning of the announcement, specifically the “surprise” part. Rather than explaining it, I will use this brilliant variant of the paradox (by R.A. Sorensen):

Exactly one of five students, Art, Bob, Carl, Don, and Eric, is to be given an exam. The teacher lines them up alphabetically so that each student can see the backs of the students ahead of him in alphabetical order but not the students after him. The students are shown four silver stars and one gold star. Then one star is secretly put on the back of each student. The teacher announces that the gold star is on the back of the student who must take the exam, and that that student will be surprised in the sense that he will not know he has been designated until they break formation. The students argue that this is impossible; Eric cannot be designated because if he were he would see four silver stars and would know that he was designated. The rest of the argument proceeds in the familiar way.

 

I could not possibly come up with a better example. Not only does it highlight the subtle, different meanings of “surprise”, but more importantly the absurdity of the chained argument when “surprise” is defined properly.  This is what you call an elucidating example – it makes a difficult concept easy to understand. An elucidating example is something that cannot be faked; it requires a deep understanding, good imagination, and effective communication.

This problem shows the importance of examining the premises very carefully, and clarifying each point of the problem.  It also shows that there are multiple approaches to the same problem.  Before answering a question, make sure that the problem is not in the question itself, as an ambiguous question will only lead to ambiguous answers.  Far more important than settling for an answer that seems right, is the willingness to examine and pursue a better answer.  Not all questions are answerable in life, but to quote Richard Feynman, “I’d rather have questions that cannot be answered, than have answers that cannot be questioned”.