All Questions
Tagged with polygons combinatorics
66
questions
7
votes
0
answers
372
views
When is it possible to find a regular $k$-gon in a centered $n$-gon?
For $n \geq 3$, say that a centered $n$-gon with $L$ layers is given by the origin, $(0,0)$ together with the points $$
\left\{\alpha\zeta_n^j + \beta\zeta_n^{j+1}\ \middle\vert\ 0 \leq j < n, 1 \...
- 5,072
3
votes
2
answers
140
views
Combinatorics Problem: Building numbers from the difference of the first 2021 numbers
I recently found this problem, it was part of a regional qualifier in Southern America (Venezuela- I believe) in January 2021. As I can’t find the solution anywhere and it is very different from the ...
user901311
0
votes
1
answer
788
views
How many obtuse angle triangles are possible in a regular Heptagon by joining its vertices?
I am only able to make one possible case,
Where we take any 3-consecutive vertices, since one of the vertices contains angle of Heptagon, which is approximately 128.57°, we get 7 such triangles.
I am ...
- 19
8
votes
2
answers
541
views
Number of points chosen form a polygon to have no isosceles and equilateral triangles.
Let $\Omega$ be a regular polygon with $n$ sides. Let's choose $\Gamma$ a set of vertices, for which any triangle with the vertices in $\Gamma$ is neither isosceles, nor equilateral. Find $\max |\...
0
votes
1
answer
167
views
Polytope of cone over a rational normal curve
Consider a rational normal curve $C\subset\mathbb{P}^d$ of degree $d$, and let $W\subset\mathbb{P}^{d+1}$ be a cone over $C$.
Since $C$ is a toric variety $W$ is toric as well. I would like to ask ...
- 838
3
votes
2
answers
577
views
Catalan numbers in polygons
I'm stuck on such problem: triangulation of the $n$-gon is division of said $n$-gon into $(n-2)$ triangles whose sides are either sides of the $n$-gon or certain non-intersecting diagonals. How many ...
- 49
2
votes
2
answers
193
views
Binomial identity of alternating sum of products of binomial coefficients taken two at a time
I came across this identity a while back and wasn't able to prove it.
$$\sum_{i=1}^{n-3}\frac{\binom{n-3}{i}\binom{n+i-1}{i}}{i+1}\cdot(-1)^{i+1}=
\begin{cases}
0& \text{if $n$ is odd,}\\
2& \...
- 45
1
vote
1
answer
332
views
Polygon Diagonal Combinatorics
A diagonal for a polygon is defined as the line segment joining two non-adjacent points. Given an n-sided polygon, how many different diagonals can be drawn for this polygon?
I know that the number of ...
- 150
2
votes
1
answer
393
views
Number of isoceles triangles formed by the vertices of a polygon that are not equilateral
QUESTION: Let $A_1,A_2,...,A_n$ be the vertices of a regular polygon with $n$ sides. How many of the triangles $△A_iA_jA_k,1 ≤ i < j < k ≤ n,$ are isosceles but not equilateral?
MY APPROACH: ...
- 1,323
10
votes
1
answer
398
views
Intersections of circles drawn on vertices of regular polygons
Using only a compass, draw all possible circles on the vertices of a regular $n$-sided polygon.
(That is, in every ordered pair of vertices one is the center, and their distance is the radius.)
...
- 12.5k
1
vote
1
answer
251
views
Number of circles on vertices of a regular polygon
How many unique circles can we draw on vertices of a $n$-sided regular polygon? To draw a circle, pick two distinct vertices. One is the center of the circle, and the other determines the radius.
Let ...
- 12.5k
1
vote
2
answers
772
views
Schlafli symbol determining number of faces
Schlafli symbols are used to describe polytopes, but can also be used to describe more general objects through the use of flags. In particular, some information can be readily 'read-off' from a ...
- 1,903
0
votes
1
answer
178
views
Number of ways to choose a closed path of given length on a square lattice
Also known as self-avoiding polygons, this is an unsolved problem. However, to leading order in the asymptotic limit, the number of polygons of a given perimeter scales exponentially with perimeter ...
- 1,191
1
vote
1
answer
894
views
Number of right angled triangles formed by vertices of a 14-gon
Here's a question that I found on the website of International Kangaroo Maths Contest. The question goes like this:
What is the total number of right angled triangles that can be formed by joining ...
- 1,313
0
votes
1
answer
33
views
Minimum $k$ vertices that a 4 form a quadrilateral with common sides with a 2018-gon
$\mathbb{P}$ is a regular polygon with 2018 sides.
What is the minimum number $k$ of vertices that 4 of them form a convex quadrilateral with 3 common sides with $\mathbb{P}$.
My idea is to color ...
- 467