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[Phys-L] widget rate puzzle ... reasoning, scaling, et cetera



Consider the following puzzle:

It takes 5 minutes for 5 machines to make 5 widgets.
So, how many minutes does it take for 100 machines
to make 100 widgets?

This puzzle has been around for at least a decade/1/. It has
come across my desk three times in the last three weeks. It's
somewhat notorious, because if you are the least bit sloppy
or careless, you get the wrong answer. I reckon it is worth
thinking a little bit about what makes it so deceptive.

OTOH it's not worth fussing over it too much, because
it was contrived to be exceptional, and there are limits
to how much you can learn from exceptional cases.

*) I conjecture that part of the deception is that it "rhymes"
with simple "distance = rate x time" problems. It invites
reasoning by analogy to a familiar problem, even though there
is a crucial difference: There's an extra variable!

*) Similarly I conjecture that part of the deception is that
it "rhymes" with a simple scaling law. Of course if you are
the least bit systematic about formulating the scaling law
you see immediately what the problem is ... but the point
remains that the question deceives you into thinking you
can do it in your head without being systematic.

To a limited extent this illustrates my previous contention
that if a student is having a problem with scaling, it's
usually not a problem with the scaling idea itself; it's
more likely to be a deeper problem with reasoning skills.

*) It is easy to say "be more careful" and "be more skeptical"
but if you take that too far it becomes impractical. There
are a lot of truly easy problems in this world, and there are
not enough minutes in the day to permit being skeptical of
everything. The trick is to separate -- somehow -- the truly
easy problems from the booby traps.

*) Undoubtedly there's more to the story.

An interesting game/1/ is to ask this question, and at the same
time ask each student to estimate what percentage of the other
students will get it right. There is a correlation: the ones
who think it is easy tend to get it wrong.

So, I leave it to you as a question: Suppose you have a student
who gets this question wrong.
-- What do you say is the right answer?
-- Do you explain it as being similar to d = r t, only different?
-- Do you explain it in terms of scaling?
-- Can you explain how to recognize it as a booby trap?
-- Anything else?

===========================
Reference:
/1/ Shane Frederick
"Cognitive Reflection and Decision Making"
http://cbdr.cmu.edu/seminar/Frederick.pdf

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