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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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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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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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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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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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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79 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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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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51 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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31 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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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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220 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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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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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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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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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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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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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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63 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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177 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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227 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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253 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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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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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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95 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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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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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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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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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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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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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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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 ...
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Prove that every convex polygon with area $1$ is contained in a parallelogram of area $\frac{4}{3}$

Prove that every convex polygon with area $1$ is contained in a parallelogram of area $\frac{4}{3}$ I can only show that polygon is contained in a rectangle of area $2$.
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How many 100 sq km bounding boxes cover the surface are of the earth.

I need to figure out how many 100 square km bounding boxes cover the surface area of the earth. I'm trying to use the Instagram API to download data. Their API supports a lat/long with a maximum ...
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Generating M well separated points in an n-dimensional hypercube

I want to generate M n-dimensional points constrained inside a hypercube such that the points are relatively well separated. I'm playing around with this using a scripting language like R or python. ...
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Is every shape possible with a snake?

Imagine a 2d snake formed by drawing a horizontal line of length $n$. At integer points along its body, this snake can rotate its body by $90$ degrees either clockwise or counter clockwise. If we ...
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Combinatorial approach to calculate determinant

Suppose you have set of $n*n$ matrices with entries from the set $\{1,-1\}$. Then what can be the maximum determinant which you can obtain from such type of matrices.
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Minimal circle containing set of points

Suppose that there are $n$ points in the plane $x_1, x_2, \dots x_n$, and $C$ is the minimal circle (the circle with the minimal radius) that contains all of them. If there is another point $p$ ...
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there's a circle with area at least n with n+1 lattice points

Prove that we can perform a translation on a circle of area at least $n$, for $n$ being a positive integer, such that there are at least $n+1$ points enclosed or in the boundary of the circle.
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Minimum number of circles with 3 neighbors

It is possible to arrange congruent circles on the plane in such a way that no two circles overlap and each circle adjoins exactly three other circles. The picture shows an example with 16 circles. ...
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How prove $ \frac{2}{\sqrt3}F \geq s-1 $ for convex quadrilateral?

Let $Q$ be any convex quadrilateral of area $F$ and semiperimeter $s$. Suppose that length of any diagonal of $Q$ $ \geq$ length of any side of $Q$ $\geq 1$ How prove $ \frac{2}{\sqrt3}F \geq s-1 ...
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How many unit squares can overlap a given unit square without overlapping each other?

How many unit squares can overlap a given unit square without overlapping each other? @calculus has managed to arrange 7 squares (see this GeogebraTube page). This seems like the maximum ...
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Inequality on length of intervals

Let $n\ge 1$ and $\{I_j\}_{j=1}^{n}$ is a set of non-degenerate subintervals of $[0,1]$. Then show that : $$ \overline\sum \dfrac{1}{|I_j\cup I_k|}\geq n^2$$ Here $\overline\sum$ denotes ...
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Coloring $\mathbb R^n$ with $n$ colors always gives us a color with all distances.

I wanted to share a really cool but simple problem. Consider a coloring of the points of $\mathbb R^n$ with $n$ colors. Prove that there is a color $c$ such that for any $r>0$ there are two points ...