1
vote
0answers
67 views

Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{n−1})$$ (See, e.g., question The generating function for permutations ...
0
votes
0answers
50 views

How to establish this result using induction?

A point $(x,y)$ in the plane is called a lattice point if both coordinates $x$ and $y$ are integers. Let $P$ be a polygon whose vertices are lattice points. Then the area of $P$ is $I + \frac{1}{2}B ...
1
vote
1answer
39 views

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 ...
0
votes
0answers
18 views

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 ...
2
votes
2answers
33 views

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 ...
2
votes
2answers
40 views

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 $\mathbb{R}^2$ = $\binom{k}{2}$?
1
vote
1answer
25 views

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 ...
0
votes
1answer
52 views

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 ...
2
votes
2answers
73 views

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 ...
0
votes
0answers
25 views

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 ...
1
vote
2answers
54 views

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 ...
1
vote
1answer
38 views

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 $\Bbb R^3$ ...
2
votes
0answers
70 views

what is the minimum value of the angles inside these triangles?

Question: Given certain points on a square(including its sides),let these points and the verteces of the squares be the verteces of a certain number of smaller triangles, no vertices of a smaller ...
0
votes
0answers
35 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
-1
votes
1answer
93 views

2014 USAMO Problem :with Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points $$ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots $$ in the plane with the following property: For any three distinct ...
1
vote
1answer
43 views

How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
2
votes
1answer
78 views

number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
0
votes
1answer
34 views

Trying to revise a formula I was once given. How many rectangular prisms are in a $n \times n \times n$ cube?

I post it the other day. The only answer I got is that the total number of rectangular prisms in a cube is equal to ${n+1 \choose 2}^3$. But using $n=2$, I found the formula to be wrong. When counting ...
0
votes
3answers
74 views

How many square based pyramids are in a bigger pyramids?

The biggest challenge to solve the problem is that I can't really picture a pyramid. And it is hard to make a model. The pyramids I am trying to find include those on all tiers.
0
votes
1answer
47 views

How to find how many rectangular prisms ( including cubes) are in a n by n by n cube?

I somehow got the answer to be [(n+1)!/2!(n+1-2)!]^2 *n Each part of the equation represents the height, length, and width of the possible rectangular prism in the big cube. You can multiply the ...
1
vote
1answer
66 views

How to find how many cubes are in a n by n by n cube?

I tried finding the answer using combinatoric by determining how many different length and width ans height are there for a cube, given the size of the bigger cube. But the formula I got turns out not ...
2
votes
0answers
43 views

Uniqueness of projective plane of order 5

Is there a slick way to see the uniqueness of projective plane (equivalently, an affine plane) of order $5$?
3
votes
1answer
53 views

Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
2
votes
0answers
97 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
3
votes
0answers
40 views

Division of space by balls in R^n

I would like to know the generalized proof of this result: http://mathworld.wolfram.com/SpaceDivisionbySpheres.html, for $n$ dimensions. What is the maximum number of regions divided by $q$ ...
0
votes
1answer
26 views

There is no projective plane of order $10$.

I need to determine if there is a projective plane of order $10$. The Bruck-Ryser theorem tells us that if $n \equiv2, 1 \bmod 4$, and there is a projective plane of order $n$, then $n$ is a sum of ...
3
votes
1answer
80 views

Placing n points in a MxM square grid

I am facing an apparently well-known problem: placing $n$ points in a discrete grid so that the points are 'evenly' distributed. By evenly I mean that I would like the density of points to be nearly ...
7
votes
0answers
84 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
-3
votes
1answer
41 views

Maximum number of pipes used to construct the polygon

Bob has been given a Summer Vacation project by his Math's Teacher. After spending almost the whole vacation sleeping,eating,playing and doing everything (except coding), Bob is now anxious about ...
0
votes
1answer
25 views

Finding number of scalene triangles with certain perimeter?

For example, lets say you want to find the number of scalene triangles with integer side lengths such that the perimeter is less than or equal to 13. I could start listing all possible triples in ...
1
vote
1answer
31 views

intersection of lines and planes ?

if 6 lines are drawn in a plane , what is the maximum number of parts in which plane is divided by them ? If there are 'k' lines drawn in the plane then plane is divided into how many maximum parts ? ...
0
votes
1answer
17 views

Triangulation of surfaces and the number of edges in a triangulation.

I am reading a chapter on surfaces and triangulations, but I think I am losing the plot. I am reading page 650 of this book ...
0
votes
1answer
34 views

combinatorics and geometry

how many different triangles with a perimeter of 15cm can be constructed that have integral multiples of sides?? MY APPROACH : (a+b+c)=15 and three sides(...
0
votes
1answer
47 views

Number of ways to make grid

I need to construct a L x 3 grid as shown below But i can use only two shapes to make it which are : Here L is the number of small square boxes in each row. I can rotate the shapes as I want. I ...
4
votes
1answer
255 views

Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
5
votes
1answer
97 views

Intersection of lines on a plane

Suppose we have n lines on a plane such that there are $k_2$ points where two lines intersect, $k_3$ points where three lines intersect, ... , $k_n$ points where n lines intersect. How many segments ...
0
votes
1answer
35 views

Count number of triangles

If we are given N lines and out of N lines M is the set of parallel lines and we are also provided M integers m1,m2,m3.. number of parallel lines in each set. Here 2<=m1,m2,m3…<=N and ...
4
votes
1answer
70 views

Existence and unicity of a complete bounded cell in a generic hyperplane arrangement.

Let $n>d$ be integers and $H_1,\ldots,H_n$ be hyperplanes in $\mathbb{R}^d$ in generic position. By generic position I mean that if we change slightly their position, then the configuration does ...
1
vote
0answers
44 views

Maximum of the minimal distance of a set of points in an equilateral triangle

In this question, a closed triangle on a plane is a set of all points in its area and on its boundary, while an open triangle excludes its boundary. Now, the problems: Let $T$ be an equilateral ...
0
votes
1answer
393 views

How many triangle can be drawn with those points? [duplicate]

There are 7 points on the circumference of a circle.How many acute triangle can be drawn with those points. please help me to solve this problem.
1
vote
2answers
105 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
1
vote
0answers
44 views

Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
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
2answers
2k views

How to solve this cube puzzle question? [closed]

How to solve questions which is based on one main question? (This question is asked in big IT MNC's aptitude test) A cube is colored orange on one face , pink on the opposite face , brown on one face ...
1
vote
0answers
110 views

Number of classes of K-sets

I am having a plane in N dimension. Th distance between 2 points (a1,a2,...,aN) and (b1,b2,...,bN) is max{|a1-b1|, |a2-b2|, ..., |aN-bN|}. I need to to know how many K-sets exist(here K-set refers to ...
2
votes
0answers
51 views

Existence Of Congruent Triangles

Two triangles $ABC$, $XYZ$ are "good" when $AB=XY$, $AC=XZ$, $\angle ABC=\angle XYZ$. That is, when two segments are equal and a not-included angle is equal, they are "good". There are $n$ ...
4
votes
0answers
42 views

The largest regular m-gon that fits inside a regular n-gon

This question just popped into my head while doing some "for fun" math. More precisely: Let $m,n\in\Bbb{Z};m,n>2$. Let $P$ be a regular $n$-gon (let's say $P$ is the convex hull of the $n$ $n$th ...
1
vote
1answer
97 views

Cover the n-sphere with sub-hemispherical caps

Original Question (answered): Define a cap (x,Phi) to be the set of all points of the sphere that are within an angle Phi of the point x. $ 0 \le \phi < \frac{\pi}{2} $. (define the angle ...
1
vote
0answers
31 views

Symmetry group of the geometric realization of a simplicial complex

Question: Given $\Delta$ an abstract simplicial complex, can one find a geometric realization of $\Delta$ whose symmetry group is isomorphic to $Aut(\Delta)$? Relevant definitions: Let $\Delta$ an ...
1
vote
0answers
78 views

Possible combinations for a cube

When drawing one of two possible diagonals on each side of a cube, how many unique patterns are possible with regard to all sides of the cube and all possible diagonal orientations. I am stuck on ...