Find the Number of Triangles

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around finding the number of triangles in a complex geometric figure using methods other than brute force counting. Participants explore various approaches, including combinatorial reasoning, symmetry, and algorithmic counting, while addressing the challenges and nuances involved in accurately determining the count of triangles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest using a matrix to represent connections between vertices and applying an algorithm to count triangles, though this is seen as a form of brute force counting.
  • Others propose a combinatorial approach, noting that every three intersecting lines can form a triangle and discussing the implications of parallel lines on triangle formation.
  • Several participants mention the use of symmetry to simplify counting, though some express concerns about missing certain triangles due to this method.
  • There are discussions about the necessity of subtracting singular triangles, where vertices overlap, from the total count.
  • Some participants express skepticism about the effectiveness of non-brute force methods, arguing that they may complicate the problem unnecessarily.
  • There is a mention of the importance of ensuring all intersections are included in the figure for the alternative solutions to be valid.
  • Questions arise regarding the validity of counting intersections that fall outside the defined figure area as potential triangle vertices.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for counting triangles, with multiple competing views on the effectiveness of brute force counting versus combinatorial and symmetry-based approaches. The discussion remains unresolved regarding the implications of including or excluding certain intersections.

Contextual Notes

Some methods proposed depend on specific assumptions about the figure's geometry, such as the presence of all intersections and the treatment of parallel lines. There are also unresolved mathematical steps related to the counting algorithms and the implications of overlapping vertices.

  • #31
Wondering if this is how Garret Lissi got his start.

{edit}
I finally counted 160....

1767352449187.webp
 
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  • #32
My artwork isn’t as pretty but here it is anyway. Each size triangle is represented only once. There are four shapes out of 22 that are left-right symmetric wrt to the whole figure are multiplied by 4 and the rest by 8.
IMG_4659.webp
 
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  • #33
bob012345 said:
My artwork isn’t as pretty but here it is anyway. Each size triangle is represented only once. There are four shapes out of 22 that are left-right symmetric wrt to the whole figure are multiplied by 4 and the rest by 8.
...
At least you didn't screw up. I've just noticed multiple errors in my image. Glad I said it would take me a week to count all these stinkin' triangles!
 
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  • #34
One take-away from this is how rapidly a simple problem can become unmanageable. Combinatorial problems grow exponentially and the slightest modification can make the calculations very difficult. By "unmanageable", I mean it might take a supercomputer decades to solve.
 

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