Intereting Posts

How to prove that the Frobenius endomorphism is surjective?
Applications of $Ext^n$ in algebraic geometry
How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?
How to prove that $\frac1{n\cdot 2^n}\sum\limits_{k=0}^{n}k^m\binom{n}{k}\to\frac{1}{2^m}$ when $n\to\infty$
Understanding precisely the dot product…
Liouville's theorem for Banach spaces without the Hahn-Banach theorem?
Riesz's Lemma for $l^\infty$ and $\alpha = 1$
Volterra integral equation of second type
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
Convergence of $\frac{\sqrt{a_{n}}}{n}$
Finding the number of solutions to an equation?
A variation of Borel Cantelli Lemma
Show that $y = \frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive.
compact Hausdorff space and continuity
What function satisfies $F'(x) = F(2x)$?

I found this interesting question, and I was wondering if anyone could help me out.

Let P be the set of points M on the earth with the property that if you go 7 miles North from M, then 7 miles West, and finally 7 miles South, you will find yourself back at the starting point M. Is P a closed set? If not, what is the closure of P?

- Can every curve be subdivided equichordally?
- For what $n$ does a hyperbolic regular $n$-gon exist around a circle?
- Formula for solving for Cx and Cy…
- Prove that angle bisectors of a triangle are concurrent using vectors
- to find the intersection points of diagonals of a regular polygon
- Distance between triangle's centroid and incenter, given coordinates of vertices

- How can we draw a line between two distant points using a finite-length ruler?
- How would you prove that the graph of a linear equation is a straight line, and vice versa, at a “high school” level?
- Formula for solving for Cx and Cy…
- $\sqrt{A(ABCD)} =\sqrt{A(ABE)}+ \sqrt{A(CDE)}$
- Can I represent groups geometrically?
- Minkowski sum of two disks
- How to prove that Pi exists?
- How many sides does a circle have?
- What is the general equation of the ellipse that is not in the origin and rotated by an angle?
- Is it possible to solve any Euclidean geometry problem using a computer?

The “obvious” solution is the South Pole $S$. If you travel 7 miles north from $S$, it doesn’t matter how much you travel west, you’re still going to get back to $S$ once you go 7 miles south again.

The less obvious solutions are those points that **are further than 7 miles from the north pole**, and are such that after traveling 7 miles north, you lie on a latitude such that the circumference of that latitude is equal to $\frac{7}{n}$ miles, for a **natural number** $n$. If you travel 7 miles north (again, assuming that you can at all), and reach a point such that traveling 7 miles west has no ultimate effect, you will get back to where you started after going 7 miles south again.

As Mariano points out below, it would not even mean anything to “travel 7 miles north” if you are less than 7 miles from the north pole, so we must exclude these points explicitly to make sure the condition is specified for all points.

In other words, if $A_n$ is the latitude with circumference $\frac{7}{n}$, and $B_n$ is the set of points which, after traveling 7 miles north, you would reach $A_n$, then the points on $B_n$ are solutions.

$$P=\{S\}\cup\bigcup_{n\in\mathbb{N}}B_n$$

This is not a closed set; *to produce the closure*, you have to add the points that are exactly 7 miles below the North Pole – this corresponds to a circumference of 0, or “$B_\infty$”. That is,

$$\overline{P}=\{S\}\cup\left(\bigcup_{n\in\mathbb{N}}B_n\right)\cup B_\infty$$

Here is an extremely not-to-scale drawing:

- Conditional Probability Distribution
- Distribution of Max(X_i) | Min(X_i), X_i are iid uniform random variables
- A is recursive iff A is the range of an increasing function which is recursive
- How prove that $10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\le\dfrac{9}{4}$
- Counterexamples to “Naive Induction”
- Symbol for the cardinality of the continuum
- The $\gcd$ operator commutes with functions defined by linear recurrence relations
- Deriving the analytical properties of the logarithm from an algebraic definition.
- Quantifier Elimination
- Finding $\log_{-e} e$
- Mapping homotopic to the identity map has a fixed point
- Complex power of complex number
- Show that the right half-open topology on $\mathbb R$ is not metrisable.
- A Curious binomial identity
- What is the minimum $ \sigma$-algebra that contains open intervals with rational endpoints