Math Problems Vault

Problem 8.238

All three vertices of an equilateral triangle cannot be lattice points.

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Problem 8.238.1

A polygon is equiangular, has $n$ sides, and all its vertices are lattice points. Then $n$ is either 4 or 8.

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Problem 8.238.2

A regular polygon has $n$ sides and all its vertices are lattice points. Then $n=4$ .

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Problem 8.239

How thick must the trunks of the trees in a regularly spaced circular forest grow if they are to block completely the view from the center?

Let $s$ be a given positive integer. Let every lattice point $p,q$ that satisfies the inequality $1\leq p^2+q^2\leq s^2$ be the center of a circle of radius $r$ . If $r$ is sufficiently small, there exist rays running from the point 0, 0 to infinity without meeting the circles described above (the forest is transparent); such rays no longer exist if $r$ is sufficiently large (for $r=\tfrac 12$ the circles touch). Let $r=\rho$ be the value of $r$ which divides the two cases (limit of transparency). Then we have

\frac{1}{\sqrt{s^2+1}} \leq \rho < \frac1s .

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Problem 8.240

In the plane draw the widest possible infinite straight ``path'', contained between two parallels and avoiding all the lattice points, that makes an angle equal to $\arctan x$ with the $y$ -axis. Denoting the width of this path by $\varphi (x)$ , determine $f(x)=\varphi (x)\sqrt{1+x^2}$ .

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Problem 8.241

Two lattice points $x,y$ and $x',y'$ are termed congruent mod $n$ , if

x\equiv x' \pmod n , \qquad y\equiv y' \pmod n .

Among any $kn^2+1$ distinct lattice points there are always $k+1$ that are congruent mod $n$ .

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Problem 8.242

In the plane consider a domain of area (Jordan content) $F$ . It may contain no lattice points, but can always be moved parallel to itself so that $[F]+1$ lattice points lie in it.

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Problem 8.243

(1)(2)

The lattice points in the closed first quadrant are enumerable. I.e. there exist functions $f(x,y)$ defined for ordered pairs of non-negative integers $x,y$ that combine the following two properties:

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(1)

The values assumed by $f(x,y)$ form the set of numbers $1,2,3,4,5,\ldots$ .

(2)

The function $f(x,y)$ takes different values at distinct points; i.e. if $x,y,x',y'$ are integers, $x\geq 0$ , $y\geq 0$ , $x'\geq 0$ , $y'\geq 0$ , $(x-x')^2+(y-y')^2>0$ , then $f(x,y)\neq f(x',y')$ .

There even exist rational entire functions $f(x,y)$ that enumerate the lattice points, i.e. that have the properties (1) (2). An example is the function

f(x,y) = \tfrac 12(x^2+2xy+y^2+3x+y+2) = \binom {x+y+1}{2}+x+1

and the function arising from it by interchanging $x$ and $y$ [the lattice points are counted successively on the line segments

x+y=0, \quad x+y=1, \quad x+y=2, \quad \ldots , \quad x\geq 0, \quad y\geq 0 .

Let $f(x,y)$ be a rational entire function of degree $m$

f(x,y) = \varphi _0(x,y)+\varphi _1(x,y)+\cdots +\varphi _m(x,y) ,

where $\varphi_\mu (x,y)$ denotes a rational entire homogeneous function of degree $\mu$ . If $f(x,y)$ enumerates the lattice points, then clearly $\varphi_m(x,y)\geq 0$ for $x\geq 0$ , $y\geq 0$ . If $\varphi_m(x,y)>0$ for $x\geq 0$ , $y\geq 0$ , $x+y>0$ and $f(x,y)$ enumerates the lattice points, then the degree of $f(x,y)$ is

m=2 .

[The number of lattice points contained inside the contour line $f(x,y)=\text{const.}$ is related to the enclosed area.]]

Problem 8.244

Consider the square domain $\tfrac 12\leq x\leq n+\tfrac 12$ , $\tfrac 12\leq y\leq n+\tfrac 12$ as a chess board with $n^2$ squares'', i.e. by means of straight lines parallel to the axis divide it into $n^2$ squares of unit area. The problem of the $n$ queens'' requires us to choose from the $n^2$ lattice points that constitute the centers of the $n^2$ squares $n$ lattice points $(x_1,y_1),(x_2,y_2),\ldots ,(x_n,y_n)$ that satisfy the $2n(n-1)$ inequalities

x_\mu \neq x_\nu , \qquad y_\mu \neq y_\nu , \qquad x_\mu -x_\nu \neq y_\mu -y_\nu , \qquad x_\mu -x_\nu \neq -(y_\mu -y_\nu ) ,

where $\mu \neq \nu$ ; $\mu ,\nu =1,2,\ldots ,n$ . Replace the inequalities by the more restrictive ``incongruence relations''

x_\mu \not\equiv x_\nu , \qquad y_\mu \not\equiv y_\nu , \qquad x_\mu -x_\nu \not\equiv y_\mu -y_\nu , \qquad x_\mu -x_\nu \not\equiv -(y_\mu -y_\nu )

$\pmod n$ and show that these have a solution if and only if $n$ is relatively prime to 6.

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Problem 8.244.1

A regular polygon has $n$ sides and all its vertices are lattice points. Then $n$ is equal to 3, 4 or 6.

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Problem 8.244.2

(continued). Give examples showing that the three cases $n=3,4$ and 6 can actually occur.

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Problem 8.244.3

All the vertices of a regular polyhedron are lattice points. Then the polyhedron is either a cube or a tetrahedron or an octahedron.

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Problem 8.244.4

(continued). Give examples showing that the three cases mentioned above can actually occur.

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