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Re: [Phys-L] Ex: Re: triangular induction puzzle [spoiler]



I actually can’t believe I fell for this because one of my pet peeves is questions like “1, 2, 3, 4, 5, ?” Yes, there is an “obvious” answer, but I can give you a rule that will give you any number you like for the next item.

JM

On Oct 12, 2022, at 12:46 PM, John Denker via Phys-l <phys-l@mail.phys-l.org> wrote:

The problem is ill-posed.
That's the point I wanted to make.

You have to extrapolate according to some rule, and as
several people have pointed out, there are many possible
rules. Indeed, there is an infinite number of rules to
choose from. You could perhaps restrict attention to
"simple" and "elegant" rules, but even so:
a) That is an assumption that may or may not be justified.
Sometimes physics is simple and elegant, but sometimes
it is not.
b) There are a still a great many rules in that category.
The given data points are not sufficient to constrain
the answer to any particular numerical value. See below.

On 10/12/22 9:24 AM, Carl Mungan wrote:
Well, I think we shouldn't assign such a thing to our students.

The issue in my mind is the word "score" isn't clear.

IMHO we *should* teach students how to handle ill-posed
problems ... and then assign such things.

Education should prepare students to function in the
real world. Virtually all real-world problems that come
across my desk are ill-posed.

Seriously, the first thing you should do when encountering
a new problem is to ask how badly ill-posed it is.

End-of-chapter problems are typically well-posed. I find
this scandalous, because it is so unlike the real world.

Think about your own life, in the classroom and elsewhere.
When was the last time somebody said "I'm confused in one
of the following four ways. Multiple choice, pick one."
That's not how it works. Typically they ask some grossly
ill-posed question, and you have to do a whole lot of
work, going far beyond the statement of their question,
to figure out what's really going on.

=============

Here is roughly how I would grade some of the possible
answers:

a) If I have spent the last week teaching about ill-posed
problems and somebody nevertheless claims that 13 (or
any other number) is "the" right answer, I am not going
to be happy. The more confident they are of their answer,
the more unhappy I am.

b) If somebody says "It's ill-posed, i.e. underdetermined"
that is a good-enough answer.

c) If somebody says it's grossly ill-posed, such that
literally any score is possible, and then goes on to
suggest a few clearly-documented assumptions, in order
to exhibit some "typical" elements of the solution-set,
that's even better.

d) If somebody starts with (c) and extrapolates their
chosen rule(s) beyond the third figure to the fourth
figure and beyond, they get extra points for being
inquisitive. Rules that agree on the first three or
four figures may diverge wildly after that.

e) If somebody gives me a scaling law for the score (S)
as a function of the length (L) of a side, for their
chosen rule(s), I'm even more impressed. For example,
if the rule is to count all the triangles of whatever
size, it's pretty obvious that S(L) scales like L cubed,
in the large-L limit.

Some other rules that have been mentioned scale like L
squared.

This provides an interesting proof that the problem is
ill-posed. A cubic has four coefficients. All cubics
are in some sense equally "simple and elegant". The
notion that you could pin down four coefficients using
only two data points is clearly ridiculous.

Even if we make the "simple and elegant" assumption
that S(0)=0, that's still only three data points, i.e.
still not enough. You can choose S(3) equal to any
value whatsoever and still fit a cubic to all the
points.

f) If somebody coughs up a closed-form expression for
S(L), valid for all L, for some clearly-stated rule(s),
I'm yet more impressed. This requires both inquisitiveness
and persistence. For the count-triangles rule, this is
trickier than it looks.
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