1
vote
1answer
27 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
39 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 $5$?
-1
votes
0answers
31 views

Finding out the triangle numbers [on hold]

How will I find the number of errors without counting the triangle??? DO i have to find out the points and then permutations?
-1
votes
0answers
14 views

Suppose you have a polygon with $n$ vertices. How many triangulations are possible, when a 'hole' is in the middle?

Given a polygon with $n$ vertices. We know the answer to the questions "How many triangulations are there?" is Catalan numbers. However, I wish to consider a variant of this case. Suppose still that ...
3
votes
1answer
36 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
73 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
36 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
24 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
59 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 ...
6
votes
0answers
62 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
40 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
18 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 ...
0
votes
1answer
24 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
16 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
43 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 ...
3
votes
1answer
142 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
73 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
26 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
69 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
43 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
139 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
86 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
40 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 ...
0
votes
0answers
20 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
811 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
107 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
48 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
41 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
78 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
26 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
74 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 ...
3
votes
2answers
123 views

Possible sides of and octahedron

What number of unique patterns can be made if all sides of an equilateral octahedron is blue or green? How do you solve such a problem? I have only tried to solve this by a hands-on approach, i.e. ...
1
vote
2answers
259 views

Cutting a hexagon to make an equilateral triangle

The problem is to cut a regular hexagon into parts that can be put together (without overlaps or wasting any parts) to make an equilateral triangle. The cuts should all be straight. What is the ...
0
votes
0answers
42 views

A Survey on Rota's Conjecture

I am looking for a survey on Rota's Conjecture (1970). I am more interested in geometric aspects of the conjecture and any geometric content related to the conjecture. Any reference would be helpful. ...
0
votes
1answer
60 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
1
vote
0answers
84 views

Is there such an example?

Is there an example of a sequence of point sets $\left\{ S_{n}\right\} _{n=1}^{\infty}$in which $S_{n}$ is a set of $n$ points inside the unit triangle, such that the minimum altitude of the triangles ...
0
votes
1answer
51 views

Find the number of possible triangles

An interview question. We are given three positive integers p, q, r such that: p + q + r = 27 and p<q<r. Find the number of triangles that are possible ...
4
votes
1answer
240 views

How prove this$\frac{1}{P_{0}P_{1}}+\frac{1}{P_{0}P_{2}}+\cdots+\frac{1}{P_{0}P_{n}}<\sqrt{15n}$

Let $P_{0},P_{1},P_{2},\cdots,P_{n}$ be $n+1$ points in the plane. Let $ d=1$ denote the minimal value of all the distances between any two points. Prove that ...
1
vote
3answers
178 views

Integral points on a circle

Given radius $r$ which is an integer and center $(0,0)$, find the number of integral points on the circumference of the circle.
3
votes
1answer
81 views

How many triangles are there in each “layer” of Poincaré disk?

Assume we grow from a single triangle layer by layer to get the whole disk. Every time a new ring of triangles makes all the vertices of the triangles already in the picture surround by seven ...
0
votes
1answer
32 views

What is this total length

What is the value of the total length of all the edges connecting the vertices of a regular $k$-gon that is inscribed on a unit circle?
2
votes
1answer
40 views

Would this be bounded

Let $a_{m}$ be supremum of the minimum of the angle between the line segments between any $m$ points, in which the supremum is taken over all configurations of $m$ points. Is $\sqrt{m}a_{m}$ bounded ...
5
votes
1answer
83 views

Is this bounded

Let $d_{k}$ be supremum of the minimum of the pairwise distances between any $k$ points in the unit square. Is $kd_{k}$ bounded as $k\rightarrow\infty$ ?
7
votes
1answer
227 views

What is this supremum

For any $10$ points in the unit circle, what is the value of the supremum of the sum of the pairwise distances between the $10$ points, in which the supremum is taken over all configurations of 10 ...
0
votes
0answers
35 views

How to show this [duplicate]

Given $15$ lines in the plane, can anyone show that there are at least $3$ of the lines, such that the angle between any two of them is less than $\frac{\pi}{4}$?
0
votes
1answer
105 views

Why is this lemma true?

In Lemma 2 of this paper, which states that, given $l$ lines in the plane, and given $\delta > 0$, there are at least $l/( [\delta^{-1}] +1)$ of the lines, such that the angle between any two of ...
14
votes
3answers
298 views

On a Putnam's 2009 problem [duplicate]

Find all even natural numbers $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + ...
0
votes
1answer
84 views

Question related to Desargues' Theorem

The diagram below is one way of drawing two triangles ($\Delta PQR,\ \Delta P'Q'R'$) perspective from a point ($O$), with pairs of corresponding sides meeting at $D, E, F$ as in Desargues' Theorem ...
8
votes
2answers
278 views

On the problem 1 of Putnam 2009

(This is adapted from problem 1 of Putnam 2009) Find all values of $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any ...