Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this [...]]]>

Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this puzzle. That point is:

Just because something is phrased a certain way is not reason to assume that that phrasing is the best way to represent the problem. And one way to critically examine the situation is to reframe it in a mathematical equation.

Khaled said, “Interesting how, once you assume that you can implicitly trust a given source, you can be led through any logic, or illogic, and have a lot of trouble pulling yourself back to a critical mindset.”

How true. Then Mark gave a good method to understand how to see where the paradox lies when he said, “I started to write an equation, because properly written equations can solve all counting problems, but then realized that this was pointless, because adding 2 dollars to the 27 dollars the guests paid did not reflect what happened.”

Exactly! The question was phrased to lead you to believe that because the facts were a certain way (which it accurately represented) you had to see it in a certain way (which was anything but accurate).

This kind of paradox is harder to identify than simply by “fact checking.” If you do a diligent fact-check of the problem, you’ll find that no facts are misstated. In fact, everything in the entire problem is on the up-and-up, except for the last sentence – “Adding the two dollars that the bellboy kept would make a total of $29 dollars.”

There’s the rub. Why would you add the two dollars that the bellboy kept?

A good way to look at the puzzle is to “follow the money,” or mentally picture the flow of what went where, instead of just listening to the arguer’s “logic” and being lead down the garden path.

So let’s follow the money:
$30 went from the men to the manager.
$5 went to the bellboy.
Of that $5, $3 went to the men, and $2 was kept by the bellboy. There is no reason to add two dollars to anything.

The last sentence of the puzzle is added just to throw you off the actual path of the money. Magician’s do this all the time. It’s called “misdirection.” but when magicians do it, they (hopefully) are doing it for entertainment purposes, only.

Magicians thrive on what we call, “the willing suspension of disbelief.” We assume you came to the show to relax your mind and just have fun in the fantasy world of “that which can’t and does.”

On the other hand, politicians, fanatical religious lunatics, some salesmen, and an awful lot of educational policy-makers thrive on the “the unwitting, or coerced suspension of disbelief.” And that makes all the difference.

If you willingly part with something that is not inalienable – (your temporary suspension of belief, your money, etc.) – well, that is your decision. On the other hand, if someone coerces you or tricks you into actually accepting something as real, or takes your money without your agreement (as in a sale, loan, etc.) they are committing a crime.

And stealing your mind is a lot worse than stealing your money, in the long run.

That is one of the reasons I started Math Mojo to begin with.
Quick story:

I used to work in a Job Corps facility. I won’t go into detail, but in a nutshell, Job Corps is a government boondoggle set up to have corporations get money for running educational and vocational programs for deserving sixteen through eighteen year-olds who have been shafted by the traditional system, or their neighborhood, parents, etc. In reality, Job Corps shafts these kids pretty badly, as well.

At the Job Corps, I was a math teacher. Basically, they wanted me to administer cheap computer-generated quizzes covering basic “math facts.” The system was so dismal I cried many nights working past midnight at my desk to try to fix it even minimally.

OK – to the point – there was one female student who was very intelligent and mature for her age, but who’d been hopelessly victimized by her upbringing. She still had some very bad vestiges of the “we’re just victims” syndrome.

One day she came to me and said, conspiratorially, “You know, Mr. Foley, we all have electronic chips planted in our hands, so the government can track us. You know about that, don’t you?”

Oh, man! My heart fell. The best I could come up with was, “Look, you’re a poor kid from the hood, and I’m just a poor schmuck working in a rural government facility for $12 an hour. Why in the world would anyone want to track you or me? Or anyone else in this godforsaken place?”

I had thought I was doing so much good trying to teach these kids some logic, and rational ways to deal with their world through math, but still the walls had been been built so high and wide by their social backgrounds.

That girl was a nice, smart, valuable person. It DRIVES ME NUTS that our society accepts deception, abuse, and coercion of thought by politicians and educational policy-makers. ESPECIALLY by educational policy-makers, who after all, should be the front line against mind-abuse and enforced stupidity. People like that young woman should be nurtured and encouraged, not “kept down” and “inculcated.”

Back to our puzzle. They type of misdirection used in The Case of the Missing Dollar (?) has several names and versions. One of them is “Red Herring.” That will be the focus of the next post here at The Math Mojo Chronicles.

As Mark pointed out in his comment, one great way is to make an equation, which is a “schematic” of the problem, using numbers.

Another was is the method that is used by the subject of Alexander Luria’s (the great neuropsychologist) book, “Mind of a Mnemonist.” A mnemonist is a person who has a phenomenal memory. I don’t mean like your friend who knows baseball statistics. I mean like a person who memorizes every step he takes and can tell you what he ate on September 3, 1966.

It turns out that the subject, “S,” who was the mnemonist, had very interesting ways to look at math problems, too. He didn’t use symbols for numbers. He used graphic images of the situation. Here’s a taste, from page 105:

When faced with this problem:

The price of a notebook is 4 times that of a pencil. The pencil is 30 kopeks cheaper than the notebook. How much is each?

(You may want to ponder how you would solve this yourself before you read on.)

He pictured the notebook and four pencils next to it, with an equal sign between them.

Then he pictured the pencil = 30 kopeks cheaper than the notebook like this: A notebook, then an equal sign, then a pencil and a plus sign and 30 kopeks next to it.

That immediately lead him to see three pencils, an equals sign, and thirty kopeks.

From there it is easy to see that a pencil is 10 kopeks, which makes it easy to see that the answer to the problem is that each pencil costs ten kopeks, and a notebook costs forty kopeks.

To wrap this up:
You are in control of your mind. The more tools you have to solve problems with it, the less you are at the mercy of people who would like to steal and mislead your attention. And the more you are free to explore the world in ways that are beneficial to you.

You can find a very thorough and interesting (and totally understandable) discussion of the motel puzzle at this really interesting blog. Make sure you look at the clever post by walldog in the comments

Another good resource to understand the motel puzzle can be found at Wikipedia.

Original Photo by Norsehorse Edited by Brian

There’s a braintwister that’s been going around the internet, well, probably ever since there was an internet. It’s actually probably thousands of years old in one version or another. You may have seen it phrased like this:

Three men go into a motel. The man behind [...]]]>

Original Photo by Norsehorse Edited by Brian

There’s a braintwister that’s been going around the internet, well, probably ever since there was an internet. It’s actually probably thousands of years old in one version or another. You may have seen it phrased like this:

Three men go into a motel. The man behind the desk said that the room costs $30. So each man paid $10 and went to the room.

Later, the desk clerk realized that the room was only $25. So he sent the bellboy to the men’s rooms with five one-dollar bills.

The bellboy couldn’t figure out how to split five dollars evenly three ways, so he gave each man one dollar, and kept the other two for himself.

This meant that the three men had each paid $9 for their rooms, which makes a total of $27 dollars. Adding the two dollars that the bellboy kept would make a total of $29 dollars.

So where is the other dollar?

My advice to anyone trying to solve anything like this, or trying to think about anything at all, for that matter, is not to jump to conclusions.

Want to give it a try and add your thoughts in a comment? Go for it! I’m not asking for the solution, just some thoughts about the meaning of the puzzle – how it relates to life, logic, decision-making and understanding your world. I am not putting this up as a trivial puzzle.

My comments will be in the next post.

(Note: When I originally posted this, there were a few typos and other mistakes in it. If you busted your head over it till now, please accept my apologies. It should be correct now.)

“When playing this simple game, people consistently reject the rational choice. In fact, by acting [...]]]> This post is concerned with a very interesting problem, called “The Traveler’s Dilemma.” There is a very good article about it, written by it’s creator, Professor Kaushik Basu, in the June, 2007 issue of the Scientific American. The article begins:

“When playing this simple game, people consistently reject the rational choice. In fact, by acting illogically, they end up reaping a larger reward–an outcome that demands a new kind of formal reasoning.”

Please read the article before you read this post.


I want to preface this post by saying that I am fascinated with the Idea of game-theory, and I admire it’s students and professors. I am in no way “officially qualified” to gainsay anything by game-theorists, or by “real” mathematicians or economists.

On the other hand, a little nudge from us plebeians can be good for the aristocracy.

Although the “Traveler’s Dilemma” is an endlessly fascinating subject, I don’t believe the dilemma is the one being presented. Or at least if it is, it is ill-named. Let me explain:

I believe that the problem should be called “The Game Theorist’s Dilemma”, or “The Clever Mathematician’s Dilemma,” or something like that. It certainly doesn’t have all the elements of the traveler’s dilemma. One of the most important elements was left out.

At no point in this article did I read about the actual price of the item. That is something both travelers would know, but is never given as a piece of their dilemma. Therefore the mathematician is playing a significantly different game than the travelers are.

(Note: The assumption that both have paid the same price is derived from the wording of the “Traveler’s Dilemma,” although, as with everything else, I could be wrong about this. I await your comments.)

Rather than gaming the system, their main dilemma is “should I lie or tell the truth?”

If each assumes that the other is basically honest, the game theorist can go home and play with himself. (Absolutely no disrespect intended. Game-theorists are way ahead of me academically – I just liked the sentence.)

If both Pete and Lucy are honest, there is no dilemma – both tell the truth. Game over.

If Pete is honest and Lucy assumes that Pete is not, (or vice-versa) then his or her dilemma is:
“Should I lowball or highball ‘Goody-two-shoes’?”

It seems to me the strategy would be to lowball by one dollar, thereby ending up with one dollar more than the honest price.

If both are dishonest, let the game-theory begin!

One thing is for sure, the first sentence of the last paragraph in the article is most poignant – “If I were to play this game, I would say to myself: “Forget game-theoretic logic.”

Later in the same paragraph, it goes on to state:

“What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality* attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler’s Dilemma.”

Note: In the sidebar to this article (the  Payoff Matrix of the Traveler’s Dilemma link), you can read:

“This payoff matrix summarized everything game-thorists need to know about Traveler’s Dilemma.”

What a statement! That’s only true if game-theorists are not concerned with anything beyond the narrow scope of staring at their own gamey navels. (Again, please, no disrespect intended.)

The most significant point of all, I think, was the anecdote about the Indian hat-seller. The subsequent discussion about it still, alas, does not bring up the fact that Pete and Lucy knew the price.

I find it odd that game-theorists apparently don’t look at this factor, considering how similar it is to the “Monty Hall Problem,” in which all the difference is made by the fact that Monte knows what is behind the doors.

At the end of the discussion about the Indian hat-seller, Professor Basu says,

” In my opinion, the common knowledge of rationality assumed by game theorists faces a … demise.”

*I don’t know if I am comfortable with the term “meta-rationality:” for so many reasons. I hope it doesn’t become default. But it would be a fine thing if some study evolved which dealt with these things. Game-theory has proven to be incredibly useful in the real world. Besides its utility, it is a field of endless fascination for your mind. But like the mind, (at least mine) it seems that it can use some maturing. Professor Basu seems very enlightened on this point, and I hope that means that I am on the right track.

So far, I have found nothing that approaches Edux theory. Presenting Edux theory will be the next step after Math Mojo is a completed site. Edux is the brainchild of Dr. Kent Lawson, Professor Emeritus of Physics and Edux at the State University of New York at Oneonta.

You can find more information about Mr. Kaushik at his home page, and at Wikipedia.

You will find more and more references to Edux here and at MathMojo.com. Stay tuned.

While listening to NPR, I heard an interesting story about how political candidates affect each other. You can hear [...]]]> “Common sense is the collection of prejudices acquired by age eighteen.” – Albert Einstein

Towards the end of making math more meaningful, I’d like to discuss something in recent news that resonates with that theme.

While listening to NPR, I heard an interesting story about how political candidates affect each other. You can hear a podcast of the same story here. You can also read the Washington Post’s story (which broke first).

The story concerns what is called “The Decoy Effect” or “Asymmetrical Dominance Effect” in psychology.

In simple terms, the Decoy Effect suggests that if you are faced with two popular choices, the outcome of your choice can be subtly affected by the introduction of a third, less popular choice (the decoy). But the outcome may not be affected in the way you might expect. The introduction of the third choice would have you lean towards choosing the popular choice that is most like the decoy.

The above-mentioned article concerns itself with front-running candidates for the 2008 presidential race.

In a (very simplified) example, the Decoy Effect suggests that in the case of Gore vs. Bush in 2004, the introduction of Ralph Nader as a liberal candidate did not (as generally assumed) take votes away from Gore, but actually increased the share of votes that Gore would have gotten had Nader not run.

There are several pitfalls here. You may be tempted to think, “Hmm, so my vote for Nader didn’t help Gore lose the election, after all!” That would not be a conclusion supported by the Decoy Effect. The Decoy Effect postulates that Nader running would not negatively affect Gore. But you voting for him would, of course.

Naturally, the effect doesn’t imply that no one would vote for Nader, just that many more would vote for Gore.

(Caveat – Please don’t flood me with e-mails about your personal politics. I am merely using this to explain a concept).

Here’s a scenario that might help you visualize the effect:

You’re Lou Costello and I’m Bud Abbott (or vice-versa if that’s the way you want to be about it). We’re at a carnival with our beautiful friend, Marjory. We are standing between two rides. One is a Ferris Wheel, and the other is “The PSYCHO-TERROR-CYCLONE from HELL.”

If we choose the Ferris Wheel, we think Marjory might think we are too “chicken” to go on a real scary ride. But we both know that we are probably going to blow lunch on the “P-T-C from H”. We’re going to have to tough it out, though, and hope for the best, because we both want to impress Marjory. So it looks like we may have to opt for the “P-T-C from H.”

Now along comes our friend, Stinky, and he tells that he is going on the bumper-cars.

“Bumper-cars! Are you kidding! That’s for kids! We may not be crazy enough to go on the The Psycho-Terror-Cyclone from Hell, but we’re going on the Ferris Wheel, like sensible, real men!”

Get it?

In some sense, the decoy gives you an “out.” Sort of like when you are offered some expensive product, which you want, but you really can’t afford. You are also offered something that isn’t quite what you want, but you can afford it. You may tend to go for neither.

Then you are offered what is sometimes called a “slum prize.” It’s a piece of junk for free.

The theory is, you’ll say to yourself, “No, that’s junk. I can afford something better,” and you’ll go for the thing you can afford.

(And no, none of the free downloads at Math Mojo are not slum prizes, you cynic, you!)

In some sense, the decoy makes you base your decision on a trait that is less important that what you are really interested in. In the above example, you made your choice based on price, although you are not buying something that you really want.

You might think, “But that doesn’t make sense!”

Heck, some people get mad if you show them a better way to multiply.

I can understand your feeling that it doesn’t make sense, it’s the “but” that is the problem. Why would you think that people chose what makes sense? People do not generally process complex information well. We like to oversimplify things. That fits just perfectly with the Decoy Effect.

Understanding that people oversimplify things helps you navigate your world better. Of course, you don’t want to oversimplify things yourself (although we all do it to some extent) you just don’t want to be surprised when other people do.

This tendency to oversimplify things is what we call “intuition.”

• “Of course introducing a similar candidate will hurt the original candidate’s choice.”

• “Well somebody’s got to will the lottery, so why shouldn’t it be me?”

• “If you don’t believe in God, well, then what do you believe in?”

• “If Johnny Jones jumped off the Empire State Building, would you jump off the Empire State building?”

All of those are questions, suggestions or implications that are based on meaningless logical fallacies, but they appear to be sound to people who want simple representations of complex problems.

It is very hard for most of us to look more deeply into things that seem simple on the surface. It is difficult to sort the simple from the complex. So the simplest thing to do is assume that any complex question is simpler than it appears. That keeps us from having to put effort into dealing with it. It keeps us comfortable, up to a point. And it keeps us wrong.

Laziness is no substitute for thought.

There was a time when we may have believed in Santa Claus. When our friend told us there was no Santa Claus, we didn’t believe him, because, “everybody knows, etc.”

Eventually we matured, and understood that much of the stuff we believed when we were children was just convenient nonsense. Then we became adults, and we knew “what the truth is”.

I believe that that moment is the worst moment in most people’s development. It’s the moment when we think we know enough. It’s the moment we have effectively stopped thinking and learning.

If we are very lucky, at some time we are faced with greater truths, which are less apparent than those “truths” we “were told in school,” or “read somewhere,” – insights that not everybody is aware of. Something that most people are too hung-up to even consider. It may even frighten them.

Imagine how some people felt when they saw the Wright brothers’ fly for the first time.

• “Galileo, how dare you believe that the earth revolves around the sun! Everybody knows…”

• “What? Question the President? Of course Saddam has WMDs. Rumsfeld even said he knows were they are. Do you think they would lie?”

• “What do you mean ‘nothing can travel faster than light?’ What if you held a flashlight and walked forward with it? That beam of light would travel faster than if you held the flashlight still! See! I’ve got you there, Professor!”

All of those statements sound (or sounded, at one time) perfectly logical to many adults, in the same way that Santa seems perfectly logical to a child. But at some point everyone should get over them.

Developing counter-intuition is the next step after developing what we generally call “maturity.” Society cannot advance very far until it is expected of everyone. Unfortunately, large portions of society will do anything they can to inhibit your development of counter-intuition. They feel threatened by what they do not understand.

The funny thing is that it’s not that they can’t understand. It’s that they won’t understand. They don’t want to. It’s not as easy as being comfortably dumb.

“What you don’t know can’t hurt you.” Right. That’s like the toddler playing “hide-and-seek” who covers his eyes and says, “You can’t see me!” (Except it’s not pathetic when a child does it.)

Arrested development is the hallmark of humanity, it seems.

Sometimes you can go by your intuition, but unless you have the option of seeing the pitfalls of intuition (which you can only do if you have the benefit of counter-intuition), your intuition is going to be a very poor judge of reality.

Math Mojo is about finding alternative, better ways of understanding simple mathematical principals and arithmetical operations. It is also about critical thinking skills.

There will be more and more articles about how to develop counter-intuitive thoughts as Math Mojo develops.

The conclusion of the Washington Posts’ article is this:
“Don’t let salespeople tell you what issues to care about, and don’t let candidates define one another. More simply, think for yourself and be wary if a difficult choice suddenly feels simple.”

That is also the heart and soul of the message of Math Mojo. It’s also part of its brain.