All Questions
Tagged with polygons combinatorics
66
questions
1
vote
1
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83
views
Moving between polygons drawn within a convex polygon with parts of diagonals
My question is about one problem given in last round of codeforces, pretty easy to handle it, but I do not understand the other players` solutions.
We have a convex polygon and numbers it's ...
- 663
23
votes
4
answers
93k
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How many triangles can be formed by the vertices of a regular polygon of $n$ sides?
How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle ?
I have no idea where I should start to ...
- 335
1
vote
0
answers
80
views
All polygons satisfy the "normal" property.
A fancy explanation is below, but here's an edited simpler explanation because I think the jargon makes the problem seem inaccessible. In reality this problem is super accessible and I'm sure the ...
- 31
1
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0
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67
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What is the probability to pass through $1\le m\le n$ vertices of an $n$-sided polygon after $t$ seconds?
Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and ...
- 7,299
22
votes
8
answers
3k
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Number of ways to connect sets of $k$ dots in a perfect $n$-gon
Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
- 821
2
votes
1
answer
3k
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Coloring the 6 vertices of a regular hexagon with a limited use per color
I want to solve to following two-part problem. I solved the first part but I am not sure how to start on part B.
A) How many ways are there to color the 6 vertices of a regular hexagon using 4 colors ...
- 43
2
votes
2
answers
588
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Number of $r$-sided polygons in $P$ with no common edges
We have a $n$-sided convex polygon $P$. How many $r$-sided polygons $(r<n)$, with its vertices among those of $P$, can be formed such that it has no sides (edges) in common with $P$?
I tried to ...
- 5,215
14
votes
3
answers
38k
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What is the number of intersections of diagonals in a convex equilateral polygon?
Question: [See here for definitions]. Consider an arbitrary convex regular polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which $\alpha_i$ ...
user180918
0
votes
1
answer
108
views
Intersections in polygons
I'm having troubles solving the following problem which is about combinatorics:
let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices.
Each vertices are supposed to connect ...
- 285
1
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0
answers
190
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Triangulations of the concave polygon
It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
- 11
3
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1
answer
185
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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 $\sum_{i=1}^{n}\...
- 1,568
1
vote
0
answers
75
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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:
So, ...
- 6,590
1
vote
0
answers
91
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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 ...
- 1,394
0
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0
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331
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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 ...
- 295
4
votes
2
answers
9k
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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?
- 345