Psychology of Reasoning: Why do some silly problems really stump people?

[gusl]

Why do some silly problems really stump people?

The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser

What about this one? What the heck are people thinking?

Three friends, bought a recorder with $30. each of them participated with $10. The shop assistance realized after they left that the recorder is $25.. He sent after them a guy with the $5. The guy gave each of the friends a dollar, so it means that each of them paid $9. Two dollars are left with the guy who took the $5. Now the three friends paid each $9, so we have now $27 adding to them the two dollars which are left with the guy. we find that we have $29.
We know that they paid $30, so where the dollar disappeared.
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[16:13:08] Gustavo Lacerda says: each of them has $1, the thief has $2, and the shop owner $25
[16:13:17] Gustavo Lacerda says: nothing disappeared
[16:16:33] Gustavo Lacerda says: " Now the three friends paid each $9, so we have now $27 adding to them the two dollars which are left with the guy. we find that we have $29."
This makes no sense. I would like to know why this argument is tempting to a non-mathematician.

Comments

[jbouwens.livejournal.com]

This kind of problems always involve a step that is non-valid, but it is cloaked such that many people do not recognize it as an invalid step.

Same thing here:
"Now the three friends paid each $9, so we have now $27".
This is correct The guys now effectively paid $27 for the recorder: The shop has $25 and the thief $2.

"adding to them the two dollars which are left with the guy. we find that we have $29."
This is the step people don't see is invalid. It is true that they effectively paid $27. It's also true the guy has $2. It's also true that 27+2=29. The fact that adding those particular $2 to those particular $27 is a completely arbitrary operation doesn't register with many people.

[gustavolacerda.livejournal.com]

so you blame it on a lack of skepticism?

[jbouwens.livejournal.com]

Not exactly. It's just lack of insight. Some people are better at making the translation from natural language to abstract description of a problem than others.

[brkvw.livejournal.com]

Exactly.

I might be able to show another good example that tricks people (less):

http://home.capecod.net/~tpanitz/ejoke/j76.htm

Bottom line though this shows up that the context of a "work day" & "day" changes from 24 to 8 and back.

So this is sort of a contextual version of the math problem.

The one I deal with the most when talking to people is their inability to recognize friction. With everything from water pressure, electrical pressure, etc.

[jbouwens.livejournal.com]

Bottom line though this shows up that the context of a "work day" & "day" changes from 24 to 8 and back.

That, and the fact EVERYTHING is calculated based on 365 days, even though at every step the number of remaining days decreases.

ie. The boss says: 365 * 30 minutes =~ 23 days, ignoring the fact that he already subtracted 104 days for weekends, and 170 days for off-work.

[gustavolacerda.livejournal.com]

i.e. they make more mistakes in their translation?

Of course, if they were skeptical by default, they wouldn't trust the results of the calculation.

[jbouwens.livejournal.com]

I think most people understand there's a mistake in there somewhere, as the result is clearly impossible. The trick is in recognizing where this mistake is. Many people suck at that.

[ataltane.livejournal.com]

To me it seems like the trick relies on people misunderstanding "paid $9 each" to mean "paid $9 each in exchange for the recorder", i.e. "paid $9 each to the shopkeeper", which isn't true, but is accepted as true since they did in fact pay $9 each. I'm sure it also plays on people's terror of dealing with genuinely complicated operations involving large restaurant bills, many empty wine bottles, and the question who-owes-who-what when it's realised that someone paid the wrong amount :)

[gustavolacerda.livejournal.com]

it seems that people think that:
total amount of money = how much the friends paid in total + how much the thief stole

whereas the correct equation is:
total price originally = how much the friends paid total after discount + how much was returned to them in discount (2)
[16:32:39] Gustavo Lacerda says: $30 = $27 + $3


maybe people make the mistake because it resembles a correct reasoning pattern (2).

a matter of direction

[http://users.livejournal.com/_greg/]

People are visualizing a flow of money TO the purchasers, and the operation of giving them the first three dollars is presented as an addition. They are supposed to have received $2 more, another addition. So now we can fool the reader by doing an addition where we should do a subtraction. If the story had stressed the three dollars as counting DOWN, and then the two dollars were presented as being added, it wouldn't fool many.

The preference for thinking in terms of additions reminds me of the official method of counting change, by starting from the cost of an item and counting up to the amount tendered, rather than the mathematically more obvious method of subtraction. I suppose that addition is much more appealing for people who don't play around with numbers.