# Tagged Questions

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### Number of outcomes of n lines in 3d space

Given $n$ lines in a 3d space what is the number of outcomes for these lines. For example: For $n = 1$ the result is $1$. For $n = 2$ the result is $4$. ( both intersect in one point, parallel, ...
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### Elementary proof of Monsky's theorem

The following curious problem was posed in our problem set for the weekly problem solving session in our institute: Prove that a square cannot be decomposed into an odd number of triangles with ...
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### Number of triangles with property

Look at the convex polygon with n vertices.No four of them lie on the same circle. How many triangles there are with the vertices in polygon's vertices such that remaining n-3 vertices lie outside of ...
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### Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?
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### Realisations of associahedra

I seem to have lost the reference to a realisation I am interested in. Hopefully someone can steer me to a paper that fully explains the realisation. For the case $K_2$(the 5-gon) the following ...
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### Counting points in/on cuboid

Given a cuboid that extend in x,y,z axis such that |x|≤N, |y|≤N, |z|≤N where N is given and can have value up to 10^9.Now a shooter is standing at origin (0,0,0).He need to shoot on any of the ...
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### What can we say about shape of intersection area of $N$ disks on a plane?

Intersection area of two disks can be bounded by at most two arcs. Intersection area of three disks can be bounded by at most four arcs. It looks like (I'm not sure) that four disks can have common ...
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### Finding out co-linear points

How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates $(x,y)$ satisfying $1≤x≤3$ and $1≤y≤3$? It is easy that we have ...
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### How to calculate the number of lattice points in the interior and on the boundary of these figures with vertices as lattice points?

We define a point $(x,y)$ in the plane to be a lattice point if both $x$ and $y$ are integers. Now let $$S\colon= \{ (x,y) \ | \ 0 \leq x \leq m, \ 0 \leq y \leq \frac{nx}{m} \},$$ where $m$ and ...
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### Enumerating possible values of a mapping between surface and interior of sphere.

We divide a sphere of radius $R$ in $K$ identical spherical compartments. Each compartment holds a maximum of $N$ items. We define a map between a point on the sphere and the states of all ...
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### Trying to understand formula for counting regions of hyperplane arrangements in $\mathbb{R}^2$

There are up to $\binom{n}{2} + n + 1$ regions created by a hyperplane arrangement in $\mathbb{R}^2$ containing $n$ hyperplanes. I want to understand this in a demonstrative way. Each hyperplane ...
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### Why is the max. number of intersections of k lines in $\mathbb{R}^2$ = $\binom{k}{2}$?

Why is the maximum number of intersections of k lines in $\mathbb{R}^2$ = $\binom{k}{2}$?
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### Order Types, Point-line duality

I am trying to understand Order Types and their enumeration. I'm having a real hard time understanding these slides.. Especially the one I am showing. Could anyone explain to me what this slide from ...
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### Vertices that create a convex quadrilateral

In how many ways can we choose 4 vertices of a convex n-gon that create a convex quadrilateral (All the inside angles are less than 180) with at least 2 sides of the quadrilateral being sides of the ...
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### Number of lines determined given a set of points

Consider the following set of points in the $x-y$ plane: $$A=\{ (a,b)|a,b \in \mathbb{Z} \ and \ |a|+|b|\le 2\}$$ How to find the number of straight lines which pass through at least 2 points in ...
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### What is the purpose of the generalised definition of a cluster algebra?

The seeds of a cluster algebra are normally of the form $(\textbf{x},B)$ where $\textbf{x}$ is a cluster and $B$ is a skew-symmetrizable matrix. However then I have come across a more general ...
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### Binary polyhedron

I would like to propose this problem. I think that Euler's characteristic could be useful. A polyhedron with more than 8 vertices is called binary if we can assign to each vertex a number from the ...
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### Countability problem

If we have a line in a plane. Does the line have $\#\Bbb R$ points? And if yes, How many points does the plane have? More than $\#\Bbb R$ or equal to $\#\Bbb R$and why? I am talking about $\Bbb R^3$ ...