Math Puzzle - Case of the Missing Dollar(?) Part 2 (The Flip Side)

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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.

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