Answer to April/May Question – Sneaky Dog (Part 2)

What new information was given by the factory boss? At first glance, nothing. The reason we don’t see it is because the information given is indirect and subtle, but nonetheless present and important. The new information given is mutual knowledge – not merely that you know something, and someone else knows something, but you are aware that someone else ALSO knows the same thing and vice versa. Making an announcement to everyone, even if it is something that they already know, says aloud what everyone might have been guessing already and removes the doubt. The new information is letting everyone know that everyone else also knows.

Big deal, you say. The fact that I know for sure that others also know seems rather trivial – it is simply removing some lingering doubt. A big deal it is indeed. Let’s look at a well-known fairy tale of the Emperor’s New Clothes. The child’s announcement does not seem to give any new information; everyone already knows that the Emperor is naked. Yet this starts a chain reaction that does not end well for the Emperor. When the child was not subsequently reprimanded for his open assertion, it confirmed everyone’s suspicion and removed all doubt. Saying what everyone already knows can strip away the thin layer of hypocrisy that we cling dearly to and destabilize a delicate situation.

We live and thrive in a world of ambiguity, especially in Asia. It is important because of an invisible property we call “face”, which can be saved by ambiguity and plausible deniability. “Face”, at its core, is a manifestation of mutual knowledge. The skill called EQ or tact, is for saving face. It is an important social skill, as we use ambiguous statements to provide an “out”, and use allusion and implicature to our advantage. We learn the difference between “reply” and “reply to all”, and between sending a message privately and to a group chat, often the hard way. We learn that some things can be hinted at implicitly, but once explicitly said aloud cannot be unheard. We hint and suggest at what we want, from favors between friends to social relationships. We shroud our requests in linguistic twisters such as “I don’t suppose you might pass the salt” to “I was wondering if you think you could possibly let me use your car for the weekend”, which are literally absurd.

Once you understand what mutual knowledge is, you can see it in action everywhere. Naughty children in the act will actively avoid the glance of their teachers and parents to continue their rampage, because the instant their gazes meet, they can no longer claim ignorance. Kidnap victims are better off if they never see their kidnapper’s face, or if the kidnapper isn’t aware that the victim saw his face. Effective emails have the right recipients in the “cc” field, and more effective emails have recipients in the “bcc” field, albeit for different reasons. And finally, you can see why people sometimes say very strange things, such as, “I’m going to pretend that I didn’t hear that incredibly insensitive comment. So, about our project ….”

Answer to April/May Question – Sneaky Dog (Part 1)

The best way to understand why 30 dogs got kicked out on the 30th morning is to simplify. Greatly simplify.

Imagine the case where there is only one dog stealing. In this scenario, 99 villagers see one dog stealing, and the only person who doesn’t see any stealing dog realizes that the thief must be HIS dog, so he kicks out his dog the next morning.

How about the case when there are 2 dogs stealing? Well, 98 villagers see 2 dogs stealing, and 2 villagers see 1 dog stealing. After the first morning, no dog gets kicked out. That night, the two villagers both realize that since they only see one dog stealing and no dog got kicked out, the only scenario that makes sense is that their own dog must be stealing. So the 2nd morning, both kick out their dog out of the village.

The case of 3 dogs stealing goes the same way. 97 villagers see 3 dogs stealing, and 3 villagers see 2 dogs stealing. After the 2nd morning passes with no action, the 3 villagers are now aware that their dog is stealing, because otherwise 2 dogs would have been kicked out (see case above). So the 3rd morning, 3 dogs get kicked out.

You can see that the logic continues in the same fashion for 30 dogs. On the 29th morning, after no dogs are kicked out, 30 villagers are now sure that their dogs are stealing, and kick them out of the village on the 30th morning.

That was not an easy question, but hopefully the explanation above makes sense. A much more difficult question remains: What additional information was given by the factory boss to solve this problem?

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”.

February/March Question – Unexpected Hanging

To make up for last month, I am posing a question that is both humorous and thought provoking at the same time.  There are multiple ways to approach this problem, and it is surprisingly difficult to articulate on exactly what is amiss.  Good luck everyone, here is the problem:

Judge Wright has a reputation for always being right. Standing before him is a condemned prisoner, who turns out to be a logician. Judge Wright decides to have some fun with him, and says, “You will be hanged at noon on one day this week, and it will come as a surprise to you. You will not know until the executioner comes knocking on your cell door at 11:55am the day of the execution. It is 4pm Sunday now, so it will happen by noon on Friday at the latest.”

The prisoner carefully considers Judge Wright’s comments. He reasons that Friday cannot be the day of execution, because being the last possible day of execution, what kind of surprise would that be? That rules Friday out completely. How about Thursday? Well, Friday is out, so Thursday is now the last possible day of execution. But again, it wouldn’t come as a surprise either. So Thursday is also out. By similar reasoning, Wednesday, Tuesday, and Monday are all logically ruled out. The only conclusion the prisoner could come to, is that Judge Wright made a rare mistake, and he will not be hanged at all.

At 11:55am Thursday, the executioner came knocking on the cell door. And sure enough, it came as a total surprise to the prisoner. Judge Wright had been right all along 🙂

Question: What is wrong with the prisoner’s reasoning?

*** Warning: It may seem like a frivolous question, but rest assured it is not as easy as it may seem ***

Answer to January Question – Zipper around the World

The answer, surprisingly, is 6.28 meters, or 0.000016%. The additional zipper needed is proportional to the diameter increase, which is 2 meters (one on each side). 2 meters times pi is 6.28 meters.

In equation format:

Let D be the diameter of Earth at the equator in meters.

 

It is highly counterintuitive that so little is needed for such a seemingly big job.  The three choices provided were there to mislead you. Even though the range of choices I offered seemed reasonable (1%/10%/100%), sometimes the real answer is closer to “none of the above”. Always consider alternatives – real life issues often have solutions that are “none of the above”.  Indeed, real life issues often don’t have neat solutions at all.

January Question – Zipper around the World

I make zippers for a living. Every few months we produce enough zipper to go around the world, which is 40,075.16 kilometers at the equator.

One day I’m bored. I haul out 40,075,160 meters of zipper (over 700 tons) from the warehouse, and make a perfect, tight wrap around the equator.

I smile in satisfaction. Upon inspecting my work, I find that the zipper is on the ground and on the surface of the ocean, getting dirty and wet, which is obviously unacceptable. Being the fickle person I am, I decide to haul in more zipper from the warehouse, and raise the entire zipper loop one meter above ground level (and sea level).

 

The math is relatively straightforward.  However, before you work out the answer with some pencil and paper, based purely on your intuition, estimate roughly how much more zipper I need.

To help out, here is a quick guide:

1%: 400 kilometers

10%: 4,000 kilometers

100%: 40,000 kilometers

You have 10 seconds to make a guess. Ready?

 

 

 

Note: Illustrations from WHAT IF?: Serious Scientific Answers to Absurd Hypothetical Questions by Randall Munroe. Copyright©2014 by xkcd Inc. Used by permission of Houghton Mifflin Harcourt Publishing Company. All rights reserved.

 

Afterword: As you can see, I have taken the time to ask for permission from the publisher before using someone else’s work, even if they are only stick figures drawings. Sometimes all you need to do is ask (click for full image).

Answer to December Question – Man in the Mirror

The first, and rather difficult step, is to realize that the mirror doesn’t care about direction, and realizing that the problem is with your brain, not the mirror or anything else. To illustrate this, point to the right, and the image will point in the same direction. Point up, same thing. However, point towards the mirror, and the images points back at you, in the opposite direction. The key observation is that the mirror inverts not in the left/right or up/down, but the front/back direction.

The second step, is to realize what you are actually seeing in the mirror. Imagine a cone pointed towards a wall. As you push the cone into the wall, imagine a cone growing on the other side of the wall, growing as you push, in the opposite direction. You end up with a cone, pointing towards you, on the other side of the wall. Now put a red dot on the left side of the cone and a blue dot on the right side of the cone, and do the same thing. You end up with an inverted cone on the other side of the wall, with a red dot and a blue dot, on the same corresponding side. Now, imagine a human face being pushed through the wall nose first, just like the cone, with the colored dots as eyes. You end up with an image of a face on the other side, inverted front to back. The left eye is still the left eye, just flipped front to back. Although highly counterintuitive, that is the correct interpretation of the image in the mirror. Still having problems? Another way to look at it, is to take a latex glove and put it on your right hand. Now, take off the glove by inverting it, so it is inside out. The inverted right-hand glove is the analog of the image in the mirror, even though it looks like a left-hand glove.

The question now becomes, why do we so instinctively see a person swapped in the left/right direction, to the point where you cannot help but see it that way? The reason, simply put, is that it requires the least work from the brain. The correct interpretation (inverting), requires an incredible amount of work, as evidenced by the effort it takes simply to imagine it. There is no existing brain circuitry to do an inversion, because there was no need to do so when the brain evolved. It is far easier for the brain to treat the image as “someone” facing you rather than an inverted meaningless image. The agent detection circuitry in your brain is where the problem is, not the mirror.

Once the brain treats it as a “person”, it needs to orient the “person” in space to make sense of it. There are two main ways to mentally turn objects around in space, around a vertical axis (turning around), or around a horizontal axis (think foosball). Technically speaking both are equally valid (as are any diagonal axes), and our brain will use the existing evolved circuitry, which is to turn around the vertical axis and spin the “person” around to face you. Interestingly enough, if one were to mentally flip around the horizontal axis, one would see the image as flipped in the up/down but not left/right direction, further showing that the problem arises from a hardwired preference in your brain.

This example shows that something seemingly so real and veridical, is no more than an erroneous representation concocted by the brain. The explanation is readily verifiable and probably enough to change your mind, even though it is counterintuitive and doesn’t “feel” right. More importantly, it should deeply challenge our beliefs about how we acquire knowledge and its validity; after all, if something that seems so real and taken for granted is in fact just an illusion, what about knowledge acquired on much shakier ground?  Another salient example is #dressgate, in which people are absolutely convinced that the dress they see with their own eyes, is either white and gold, or blue and black.  It is almost inconceivable that others may see it entirely differently.

Bertrand Russell once said “The fundamental cause of trouble in the world today is that the stupid are cocksure while the intelligent are full of doubt”. Undeniably, the will to doubt requires far more than the will to believe. Doubting others is comparatively easy; far more difficult is the will to doubt yourself and carefully examine even your most cherished and emotionally invested beliefs.

“Some things you need to see to believe; some things you need to believe to see”

December Question – Man in the Mirror

It’s holiday season. After a full year’s work and study, we are all anxious for this well-deserved break. In addition to giving thanks, sending and receiving gifts, and participating in holiday activities, it is also time to look back at the past year and reflect.

This month’s question, is about reflection. We look in the mirror every day, and see our own image looking right back at us. However, something is a bit different. When I lift my right hand, the mirror image appears to lift his left hand. So left and right seem to be flipped. Yet, top and bottom are not flipped at all. If you think carefully about it, it should make absolutely no sense, because a mirror is a piece of glass that should treat left/right and top/bottom exactly the same way.

So why does a mirror image seem flipped in the left/right, but not the top/bottom direction? It’s a simple question, but not an easy question. Good luck and happy holidays.

 

Images provided by Classroom Clipart

Answer to November Question – Daily Lottery

If you chose 5) There is no difference, every day is equally as likely, congratulations, you have good statistical sense, and are probably quite sure of your answer, but you are nonetheless wrong – just less wrong. The correct answer is 1), tomorrow.

Counterintuitive? Here’s why.

Yes, every day is equally as likely to hit the lottery. Namely, a 1% chance.  So what is wrong?

Let’s look at the actual question.  The question was not “what day am I most likely to win the lottery?” –  it was “what day am I most likely to stop playing?”.  At first glance the two questions seem to be the same, yet there is a subtle but very important difference.

Let’s say today is Sunday. Tomorrow (Monday) I have a 1% chance to win. The day after tomorrow (Tuesday) also carries a 1% chance to win. The same goes for Wednesday, and every day after that. However, while the chance I win on Tuesday is 1%, the chance that I stop playing on Tuesday is less than that, because I must not have already won on Monday.  If I had gotten lucky on Monday and won, I wouldn’t even have had a chance to play on Tuesday, because I would have stopped already.

Each day after that, the chance that I stop on that particular day decreases accordingly, not because I’m less likely to win on that day, but because I cannot have already won any day before then. Therefore, the most likely day that I will end up stop playing is tomorrow, which carries a 1% chance. Every day after that carries a chance of less than 1%.

This is a good example of how our intuitions fail us. I stated clearly that it is not a trick question, but “you just need to understand the question”, and for good reason. The question asked was “what day am I most likely to stop playing?”, which many people immediately substituted for a much easier question, “what day am I most likely to win?”. There is a subtle but important difference, which is the hard-to-spot implied condition of previous losses. To stop playing on a day does not just mean you win that day, but also that you cannot have won before that day.

The last option “There is no difference, every day is equally as likely” is so appealing because it is a true statement.  The statement just happens to be irrelevant to the question. It is a red herring to throw you off the trail.  Similar to a mental sleight of hand, it’s a powerful technique, widely used by marketers, politicians, monthly question askers, and boyfriends/girlfriends.