Combinatorial geometry is concerned with combinatorial properties and constructive methods of discrete geometric objects. Questions on this topic are on packing, covering, coloring, folding, symmetry, tiling, partitioning, decomposition, and illumination problems.

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Combinatorial problem of choosing points inside an equilateral triangle without them being too close.

Determine the smallest integer $m_n$ which satisfies the following property: If $m_n$ points are chosen inside an equilateral triangle of sides 1, then at least two of them are at distance ...
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66 views

Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
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Existence of fair parallel queues

I just spent a few days at a major theme park. The queue for one particular ride (involving pirates) bifurcated upon entry; the two sides wound independently through a maze and emerged next to each ...
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2answers
50 views

Self-avoiding rook walks on small rectangular chessboards (contest question)

I am not sure how to get a closed-form formula for $R(3,n)$ as the recursion involves a summation. Maybe the best that can be achieved is a recursion that does not involve a summation having an upper ...
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1answer
35 views

Intersection of Random Subspace and Hypercube

Suppose that $A \subset \mathbb{R}^n$ is a random linear subspace of dimension $k < n$. I am interested in the event that $A$ intersects the hypercube $[-1,\ 1]^n$ at a specific face. In other ...
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Building a box from smaller boxes

John has 77 boxes each having dimensions 3x3x1. Is it possible for John to build one big box with dimensions 7x9x11? I'm leaning towards no, but I would like others opinion.
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1answer
257 views

Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
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Number of triangle that can be formed from l,m,n set of parallel lines.

If we are given $\{l,m,n\}$ set of parallel lines how many triangle can be formed from them after they intersect.
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1answer
61 views

sample variance of regular polygon upon superimposition of vertices

Given, the vertices of a regular polygon, the centroid here would be the sample mean of the vertices and we assume it to be at the origin. The distance from each vertex to centroid is ...
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4answers
188 views

A circle with $500$ points in its interior

Given any $1000$ points in the plane, show that there is a circle which contains exactly $500$ of the points in its interior, and none on its circumference. How do I approach this problem? I feel ...
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46 views

Countably Infinitely Many Points in a Euclidean Space

Do there exist $d\in\mathbb{N}$ such that there are pairwise distinct points $x_1$, $y_1$, $x_2$, $y_2$, $\ldots$ in $\mathbb{R}^d$ such that (i) $\left\|x_i-y_i\right\|_2 >1$ for ...
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Configurations of eleven (or more) points in the Euclidean plane, such that out of any four there is a pair at unit distance.

Inspired by this question, I was wondering the following: What is the maximal size of a subset $C$ of the Euclidean plane such that out of any four points in $C$ there are two at unit distance ...
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1answer
30 views

The number of ways to draw boundaries of constituencies, subject to constraints

A state comprises 45 counties arranged as 5 rows, running east and west, of 9 counties each, the nine colums of 5 running north and south. They're connected horizontally and vertically, i.e. ...
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25 views

Proof for formula that uses recursion to find combinations.

In a workbook, there this formula: Number of Combinations = $(m-1)[(m-1)^{n-1}+(-1)^n]$. This formula found the number of n-pointed stars colored with m colors without having adjacent points be of the ...
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30 views

Maximum number of edges in a subgraph of hypercube

Let $H_n$ is an $n$-dimensional hypercube, $|V(H_n)|=2^n, |E(H_n)|=n2^{n-1}$. Let $M\subset V(H_n), |M|=2^k, 1\le k<n$, and $G_M$ is a subgraph of $H_n$ induced by $M$, $V(G_M)=2^k$. Prove that ...
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1answer
71 views

Is there a mathematical way to determine a solution for puzzle games?

Or more specifically, a mathematical way to determine HOW to solve a puzzle game. Take a look at this screenshot of a puzzle game "The Talos Principle." As you can see, the purpose of this ...
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0answers
35 views

An art gallery problem

An art gallery has the shape of a simple $n$-gon. Find the minimum number of watchmen needed to survey the building, no matter how complicated its shape be. I failed to solve the problem. Please help ...
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1answer
100 views

Prove: $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components.

I need to prove or at least to understand why $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components. But I've no idea how to deal with it. I even tried to draw it ...
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0answers
52 views

Fill a rectangle with squares

How many ways are there to fill a $m\times n$ rectangles with squares that have integer side lengths. Both $m$ and $n$ are integers.
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1answer
55 views

Dominoes on chessboard

A $2016\times 2016$ chessboard is tiled with $2 \times 1$ dominoes. I can prove that there is a grid line that pass through at least $505$ dominoes. But how to prove or disprove that there is a ...
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1answer
41 views

Maximal Triangle Partitioning in n lines

Recently I was given the following problem at work: Given a 5 pointed star, draw two straight lines through it so that there are 10 minimal triangles within the drawing. It took some work but I ...
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66 views

Red and blue balls lined up

On a plane, is it possible to arrange $6$ red points and $6$ blue points such that No $2$ points coincide. For any line containing two or more points, not all the points on that line are of the same ...
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Neighbor-full partition of $\{0,1\}^n$

What is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that has the maximum number of elements? (which we call $k(n)$) We say $A$ and $B$ are neighbors if their ...
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2answers
317 views

How many planar arrangements of $n$ circles?

Is there a known formula or recursion for the number of distinct arrangements of $n$ distinct circles in a plane, where two arrangements are regarded as distinct unless one can be obtained from the ...
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80 views

Is it possible to choose 10 points from 20000 with that property?

A set of $20000$ points is chosen in a ball of radius $6$ in $ \mathbb{R}^3 $. Do there exist $10$ ones of these points s. t. all the distances between them are less than or equal to $1$?
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1answer
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Hexagon Numbering Problem

So in the above hexagon figure, I have to arrange 1 to 7, inclusive, into the circles such that the three dark red triangles have the same sum. How many distinct arrangements can there be?
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3answers
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Maximum $C$ such that every shape in $\Bbb R^2$ with area $<C$ can be placed to avoid $\Bbb Z^2$

For $C=1$, it has been proved here that every shape in the plane having area less than $1$ can be translated and rotated so that it does not touch any element of $\mathbb Z^2$. (In fact, for $C=1$, ...
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1answer
37 views

Combinatorial optimization problem

I'm having trouble writing a general algorithm for what at first glance appears to be a simple problem. If I have a volume $V_{required}$ that can be made from two smaller, different volumes how can ...
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1answer
28 views

Minimum number of circular segments.

Let K be any natural number. Consider the unit square, and the circle of diameter 1 inside of the square. We then consider circular segments of area $\frac{1}{2K}$ and claim that there exists a ...
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2answers
129 views

A question about $ (2 \times 3) $-rectangles.

The following is a problem from TopCoder: Problem. Given the width and the height of a rectangular grid, return the total number of non-square rectangles that can be found on the grid. For ...
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1answer
71 views

Is there an equidissection of a unit square involving irrational coordinates?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky's theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. ...
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1answer
189 views

Number of ways to dissect a square into triangles of equal area

Monsky's theorem states that it is impossible to dissect a square into an odd number of triangles of equal area. If $n$ is an even integer, I am interested in the number of ways of dissecting a ...
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2answers
236 views

Do two closed subsets of $[0, 1]$ with measure $\frac{1}{2}$ intersect?

Let $A$ and $B$ be two closed subsets of $[0,1]$, each with a length of $1/2$. Is it always true that $A\cap B\neq \emptyset$? My intuition is yes, because: Either they intersect in their interior; ...
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1answer
291 views

Combinatorics - Integer sided triangles with integer median

The original problem states: "Given a number N, how many integer-sided triangles $(a,b,c)$ with an integer median $m_{c}$ exist, provided that $a \leq b \leq c \leq N$?". I've managed to get it down ...
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Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
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Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
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1answer
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Cardinality of a minimal open cover of the disc

Consider $D_1^2(0)=\{x\in\Bbb R^n: ||x||_2\leq 1\}$ and let $\epsilon>0$. Consider the open cover $\mathcal{O}=\{B_\epsilon^2(x):x\in D_1^2(0)\}$ of $D_1^2(0)$. What is the minimum cardinality ...
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Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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Lattice points-Triangle

We have a triangle $ T $ with vertices at the $ \mathbb{Z} \times \mathbb{Z} $ grid . Now, consider the surface $ 2T= \{x \in \mathbb{R}^2 : \frac{x}{2} \in T \} $ ( so, double $ T $ ). Is it possible ...
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76 views

Number of Points Inside a Rectangle

This question is from a Japanese contest: Let $S$ be a set of 2002 points in the coordinate plane, no two of which have the same $x$- or $y$- coordinate. For any two points $P,Q$ in $S$ consider ...
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1answer
104 views

Sum of the perimeters of the squares intersecting the main diagonal

This question is from an old Russian contest: The unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of the perimeters of all the smaller ...
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1answer
38 views

Three vertices of a n-gon [closed]

We choose three vertices of a convex n-gon, which form a triangle. If the number of ways we can choose the three vertices so that no sides of the triangle coincide with any sides of the n-gon is 7n, ...
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Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
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When is a graph a triangulation of a polygon?

This question came up in an undergraduate math club meeting yesterday. As we know, a graph is planar if it can be embedded in the plane with no edges crossing. A famous necessary and sufficient ...
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1answer
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Combinatorics question about alternately-coloured diagonal halves of sides of a cube

Diagonal halves of each side of a cube are painted in alternate colours. Let the vertex at which such a half forms a right angle be its base vertex. What is the minimum number and the maximum number ...
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1answer
81 views

To determine number of arrangements of 4 letters in a word so that the transitions remains conserved

A 10 letter word is composed of $A,\ B,\ C,\ D$. The problem is to find the number of arrangements of these alphabets which could lead to fixed number of transitions between each pair of alphabets. ...
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1answer
182 views

Points “seeing” each other in a loop

For two points $P,Q$ with integer coordinates in $2$ dimensions, we say that $P$ "sees" $Q$ iff the segment $PQ$ contains no other points with integer coordinates. Do there exist points ...
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51 views

How many regions created by lines,circles, lines and circles, ellipses, spheres, planes from cutting?

I came up with a fantastic exercise (well not so fantastic but I think its a good generalization of classic things). In each of the following cases, find (1) the maximum number of points of ...
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2answers
81 views

Placing dominoes on a chessboard

Find the smallest number of dominoes we must place on an $8×8$ chessboard, so that in every $2×2$ square, at least one of the squares is covered by a domino. I am getting confused again and again as I ...
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3answers
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Cutting and reassembling squares

Is there a general way to cut a square into polygonal pieces so that the pieces can be assembled into n equally sized squares for each n? For example, 2 and 4 and n=k^2 is obvious (2 by the diagonals ...