3
votes
1answer
68 views

Labeling the vertices of a polygon with 0's and 1's

Suppose $P_n$ is the regular polygon with n vertices ($n\geq 5$). Let $V=\{v_1,\ldots,v_n\}$ be the vertex set. I would like to define a labeling function $\ell:V\to \{0,1\}$ so that ...
0
votes
0answers
18 views

Regular polygon with $n$ sides , the number of triangles [duplicate]

For a regular polygon with $n$ sides $(n>5)$, the number of triangles whose vertices are joining non-adjacent vertices of the polygon is $n(n-4)(n-5)$. When I take $n=6$, I get David's Star: ...
1
vote
0answers
27 views

Rectilinear polygons winding around a torus

A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($ 90^{\circ}$) and that of interior concave angles ($ 270^{\circ}$) is always $4$. Consider a ...
0
votes
0answers
90 views

Solving the “Library of Babel” puzzle, but for polygons.

The Library of Babel is a story about a universe whose contents are every possible 410-page book that could possibly exist. After a conversation with someone about doing this with images, and coming ...
3
votes
2answers
1k views

Number of triangles in a regular polygon

A regular polygon with $n$ sides. Where $(n > 5)$. The number of triangles whose vertices are joining non-adjacent vertices of the polygon is?
1
vote
0answers
65 views

Corresponding Triangulations of an (n+2)-gon to n Segments Connecting n+1 Collinear Points

So I'm asked to count the number of ways of connecting n+1 collinear points with n line segments subjected to the following constraints: If the line is L 1) No segment passes below L. 2) Starting at ...
2
votes
1answer
414 views

How many triangles are formed by $n$ chords of a circle? [duplicate]

This is a homework problem I have to solve, and I think I might be misunderstanding it. I'm translating it from Polish word for word. $n$ points are placed on a circle, and all the chords whose ...
8
votes
2answers
158 views

Why does the term ${\frac{1}{n-1}} {2n-4\choose n-2}$ counts the number of possible triangulations in a polygon?

In the given picture bellow, it counts the number of different triangloations in a polygon, how do the get to this expression, why is it: $$ {2n-4\choose n-2} $$ and why do we multiply it by ...
1
vote
1answer
40 views

Find the minimal number of guard points of polygon

Given a polygon with $n$ vertices, what is the minimal number of points inside the polygon such that for each interior point there exists at least one point such that the segment between them lies ...
0
votes
2answers
58 views

Property of nonconvex polygons

How to prove that each non-convex polygon with no self-intersecting parts, has at least one interior angle which size is less then $180$ degrees.
0
votes
0answers
175 views

k-dissection of a polygon with non-intersecting diagonals

I am trying to use the vertex coalescing method like the one mentioned here, page 10, to count: Number of dissections of a polygon using non-intersecting diagonals into even number of regions. I am ...
3
votes
0answers
93 views

Polyonimo Tiling

I came up with the following conjecture the other day, and was wondering if the result was well-known or even true: Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total ...