All Questions
Tagged with polygons combinatorics
66
questions
9
votes
1
answer
460
views
Number of chords in a $n$-gon if each chord is crossed at most $k$ times
Consider an $n$-gon where we denote the points by $v_1, \dots, v_n$.
If we allow each chord (internal edge of the $n$-gon) to have at most $k$ crossings, how many chords can we put into the $n$-gon (...
- 61
6
votes
1
answer
599
views
Smallest circumscribed polygon around regular polygons
Given a regular $n$-gon $Q$, there are many polygons $P$ that entirely contain $Q$, and such that all $n$ vertices of $Q$ lie on edges of $P$. These circumscribing polygons $P$ have different numbers ...
- 63
3
votes
2
answers
64
views
Need a hint regarding this question...
In how many ways can we select three vertices from a regular polygon having $2n+1$ sides ($n>0$) such that the resulting triangle contains the centre of the polygon?
- 31
3
votes
2
answers
4k
views
Number of Quadrilaterals that can be formed in Decagon
Find number of Quadrilaterals that can be formed in a Decagon such that no side of Quadrilateral is common to side of Decagon.
I tried as follows:
Arbitrarily choose $6$ points on a circle. Then we ...
- 10.2k
2
votes
1
answer
128
views
Looking for a formula that counts the number of unique constructions of "coffee tables" that have an n-gon tabletop.
Coffee tables are build by placing legs under vertices of an n-gon in such a way that the table won't tip over. Here are the rules:
If n is even we can use (2, ..., n) legs. Every valid table must ...
- 351
2
votes
1
answer
606
views
The travelling salesman problem for a regular n-gon
The TSP asks, given a finite set $V$ of points in $\Bbb R^2$, to find the shortest path that passes through all points and returns to the starting point. Trivially, one reduces to the case of a path ...
- 27.7k
5
votes
3
answers
151
views
Lattice embeddings of a polygon
Consider the four lattice polygons below. Each shape is over the coordinates.
If reflected or flipped on the major axes and diagonals, these four polygons remain distinct. However, this is the ...
- 21.4k
0
votes
2
answers
232
views
How many common points do two regular polygons in a circle have?
I was attempting to solve the following question:
In a circle you have a $27$ sided regular polygon and a $297$ sided polygon $($all vertices are on the circle$).$ How many common points do they ...
- 1,368
2
votes
1
answer
156
views
Number of regular (not necessarily simple) polygons on $n$ equidistant points on a circle.
I need to find the number of regular $n$-polygons on $n$ equidistant points on a circle (that is, adjacent points are equally distant from each other). There's a hint saying the answer is related to ...
- 107
6
votes
0
answers
297
views
Counting Regular polygons in Complete Graphs
The figure shows the correct $24-$gon, which held all the diagonals.
a) Find out how we got right triangles and squares (question for arbitrary $n$)?
b) How this problem can be generalized (if it is ...
- 17.9k
4
votes
2
answers
5k
views
Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon
How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon?
It's quite easy to solve for triangles for the same question,...
- 199
2
votes
1
answer
107
views
Prove the number of red sides are always larger than $\frac{n^{2}-2n}{2}$
Every side and diagonal of a polygon (n-sided) is colored in red or blue. If there are no triangles with all sides colored in blue, prove that the number of red sides is always greater than $\frac{n^{...
1
vote
2
answers
265
views
number of subset forming polygon
Given a set $S = \{ 1 , 2 , 3,\ldots, n\}$. How can I find number of subsets of size $K$ ($K < n$) whose elements taken as length of edges can form a convex polygon ($K$-sided).
- 9
0
votes
1
answer
67
views
Number of Pieces a regular $n$-gon is cut into by its diagonals [closed]
In how many pieces a regular n-gon is cut into by its diagonals?
I need a general formula.
By inspection, I have the solution to some lower values of $n$.
For $n=3,4,5,6$ solutions are $1, 4, 11, ...
1
vote
0
answers
243
views
Minimal diagonal intersections in a convex polygon
OEIS A006561 gives the number of intersection points in the diagonals of a regular polygon. There's a paper by Poonen. For 4 vertices to 12, the number of intersection points is:
$$1, 5, 13, 35, 49, ...
- 21.4k