Re: [Phys-L] triangular induction puzzle

[Timothy Folkerts <folkertst@nicc.edu>]

RULE 1:  Count all the "downward sides" of the small triangles.
1st Figure:  has one triangle with one "base"  =1
The 2nd figure has three 'rightside up' triangles with 3 bases, + 1 'upside
down' triangle with 2 downward sides.
3+2 = 5
The 3rd figure has 6 "rightside up' @ 1 each + 3 upside down @ 2 each
6 + 6 = 12

RULE 2: Count all the horizontal bases in all the triangles
1st Figure:  1 horizontal segments in 1 triangle = 1
2nd Figure:  3 horizontal segments in 3 small triangles + 2 horizontal
segments in 1 medium triangle.  (The upsidedown triangle doesn't count)
3+2 = 5
3rd Figure: 6 small triangles @ 1 each + 3 medium triangles @ 2 each + 1
large triangle @ 3 each
6+6+3 = 15

RULE 3:  1.5(L^2) - 0.5L
1st Figure: 1.5(1) - 0.5(1) = 1
2nd Figure: 1.5(2^2) - 0.5(2) = 5
3rd Figure: 1.5(3^2) - 0.5(3) = 12

RULE 4: (3/14)L^4 + (11/14)L
1st: 3/14 + 11/14 = 1
2nd: 48/14 + 22/14 = 5
3rd: 243/14 + 33/14 = 19.71428...

Of course, there are infinite rules that could be proposed with an
infinite number of possible answers.

Tim F





On Wed, Oct 12, 2022 at 1:47 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.
> _______________________________________________
> Forum for Physics Educators
> Phys-l@mail.phys-l.org
> https://www.phys-l.org/mailman/listinfo/phys-l

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