Re: [Phys-L] [**External**] Re: triangular induction puzzle

[Carl Mungan <mungan@usna.edu>]

Yes, now that's the kind of puzzle I like. Counting upright triangles is
pretty straightforward and I get R(n) = n(n+1)(n+2)/6. Counting upside-down
triangles to find U(n) seems trickier and I don't immediately see how to do
it without drawing the pattern out to at least n=5 if not further.

On Wed, Oct 12, 2022 at 10:37 AM Philip Keller via Phys-l <
phys-l@mail.phys-l.org> wrote:

> Well it is a kind of game!  I am eager to hear the interpretations that
> give 1,5,12... or 1,5,14....  I just see 1,513.  Also, for fun, I just
> challenged my students with this variant: assume we are counting
> triangles.  T(1)=1.  T(2)=5.  T(3)=?  And then: T(n)=?
>
> On Wed, Oct 12, 2022 at 12:24 PM Carl Mungan via Phys-l <
> phys-l@mail.phys-l.org> 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.
> >
> > Would there be any issues if the puzzle replaced the word "score" with
> > "total number of triangles"?
> >
> > If not, it *is* a word game.
> >
> > But maybe I'm missing something. -Carl
> >
> > On Wed, Oct 12, 2022 at 12:47 AM John Denker via Phys-l <
> > phys-l@mail.phys-l.org> wrote:
> >
> > > Hi --
> > >
> > > Here is a puzzle that has been floating around recently:
> > >    https://av8n.com/physics/img48/tri-inference.png
> > >
> > > Given:
> > > The first figure has a score of 1.
> > > The second figure has a score of 5.
> > >
> > > Questions:
> > > a) What is the score of the third figure?
> > > b) How do you know?
> > > c) How sure are you?
> > >
> > > Remarks:
> > > * Hint: It's harder than it looks.
> > > * There's no physics in it per_se, but similar situations
> > >   arise in physics All The Time.
> > > * This is not a word game. No wise-guy dirty tricks. The
> > >   things that look like triangles are triangles. The things
> > >   that look to be congruent are congruent.
> > > * Imagine assigning this to your students. Think about
> > >   what you would infer from the various answers you get.
> > > * Hint: This can be used to illustrate an interesting
> > >   point, more interesting than the plain numerical answer.
> > > _______________________________________________
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> >
> >
> > --
> > Carl E. Mungan, Professor of Physics  410-293-6680 (O) -3729 (F)
> > Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
> > mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
> > _______________________________________________
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> > Phys-l@mail.phys-l.org
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> _______________________________________________
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-- 
Carl E. Mungan, Professor of Physics  410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/