# Tagged Questions

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### How to divide a $4\times 4$ square in six pieces to show that from any seven points in the square, there are two at most $\sqrt 5$ apart?

Let $R$ be a $4 \times 4$ square. For any seven points on $R$, there exists at least two of them, namely $\{A,B\}$, with $d(A,B)\le\sqrt{5}.$ (Old problem): If $R$ is a rectangular region ...
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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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### Existence objective function given optimality regions

Let $I$ and $X$ be finite, nonempty sets, and denote by $\Delta(X)$ the set of probability measures on $(X,2^X)$. Suppose that for each $i \in I$, we are given a subset $M_i \subseteq \Delta(X)$ of ...
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### Geometry textbook

I am planning to take a graduate Geometry course next semester. The preliminary syllabus does not specify any textbook but has the following descriptions: Catalog Course Description: This course ...
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### 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 ...
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### How many points you should draw in the square at least？

There is a square, which side length is $2$, To ensure there exists a triangle in the square, with an area less than $0.5$, how many points should you draw in the square at least. the goal is for all ...
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### What is the maximum number of pieces that a pizza can be cut into by 7 knife cuts? (NBHM 2005)

I am seeing this question very first time and do not know any formal way to solve it. Which part of mathematics it is related to? What is the maximum number of pieces that a pizza can be cut into by ...
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### Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
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### 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 ...
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### Unit diameter pentagons with maximum area

In the euclidean plane, if one considers the set of quadrilaterals having unit diameter (maximum distance between two points in the convex envelope), it is quite easy to give a description of the ...
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### Expected length of the shortest polygonal path connecting random points

$N$ points are selected in a uniformly distributed random way in a disk of a unit radius. Let $L(N)$ denote the expected length of the shortest polygonal path that visits each of the points at least ...
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### Cutting up a circle to make a square

We know that there is no paper-and-scissors solution to Tarski's circle-squaring problem (my six-year-old daughter told me this while eating lunch one day) but what are the closest approximations, if ...
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### Minimum number of hemispheres covering a sphere

Here is a question which seems easy but seems to have many pitfalls. If I give you an arbitrary covering of the sphere by $N$ closed hemispheres. You can pick any of the hemispheres to keep. What is ...
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### Is this curve the circumference of a circle?

Let $\Gamma$ be a single closed curve with no self-intersections on a plane which satisfies the following condition : Condition : For any distinct four points $P, Q, R, S$ on $\Gamma$, if the line ...
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### Choosing vectors on a complex sphere

Consider a complex $t$ dimensional unit sphere. Say we pick $n$ points on this with inner products in the set $\{a_1,a_2,\dots,a_r\}$ (we have $n$ inner products with value $1$). Note the set ...
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### How many circles of radius r fit in a single bigger circle of radius R?

Is there any formula to calculate how many circles of radius r fit in a single bigger circle of radius R? I'd apreciate if it didn't involve advanced math, like calculus (unless there is no other way, ...
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### 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 ...
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### Proving a point inside a triangle is no further away than the longest side divided by $\sqrt{3}$

Problem: In a triangle $T$ , all the angles are less than 90 degrees, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the ...
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### How to prove that $5$ unit hypercubes cannot be positioned to cover a unit hypersphere?

How to prove that $5$ unit hypercubes cannot be positioned to cover a unit hypersphere? The unit hypersphere has hypervolume $\frac{\pi^2}{2} \approx 4.93 \lt 5$ but it seems unlikely that it is ...
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### Is the figure the circumference of a unit circle?

A friend of mine taught me the following question. I've never heard such a strange and interesting question! Qustion: Supposing that a figure $S$, which is constituted by points, satisfies the ...
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### Circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them

I have a serious problem with this problem: Is it possible to Draw circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them !? Any help ...
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### n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
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### Orthogonal chords of compact sets

For any compact set on a plane say C does there always exist a chord in C such that its end points are orthogonal to the boundary of C (assumed smooth)
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### Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
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### Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n - 1$ corners of that hypercube and highlight all $(n - 2)$ dimensional elements that originate from each of the corners is it ...
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### What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
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### Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
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### Two cubes in unit cube

A cube of side one contains two cubes of sides $a$ and $b$ having non-overlapping interiors. How to prove the inequality $a+b \le 1?$ The same question in higher dimensions.
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### Characterisation of linearly separable points of a hypercube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. E.g. for a cube, the following 4 points (red) are not linearly ...
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### Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
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### square cake with raisins

Alice bakes a square cake, with $n$ raisins (= points). Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins. Note that a single ...
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### Geometry - Equilateral triangle covered with five circles

I have to cover an equilateral triangle (whose sides are 1m long) with 5 identical circles: what's the minimum radius of the circles?
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### Geometric solution to classic committee problem

Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ...