Returning Explorer Puzzle/Variant 2
Puzzle
An explorer walks:
He finds himself back where he started.
Where are all the places in the world where this is possible?
Solution
- $(1): \quad$ The North Pole
- $(2): \quad$ Any position $1 + \dfrac 1 {2 \pi n}$ miles from the South Pole, where $n$ can be any (strictly) positive integer.
Proof
The North Pole solution is explored (no pun intended) in Returning Explorer Puzzle, where he is specifically located, as he then shoots a polar bear.
It remains to explore the region of the South Pole.
Let $S$ denote the South Pole.
Let the explorer starts some distance $x$ from $S$, where $x$ is greater than $1$ mile.
He walks due south to a point $P$ which is $x - 1$ miles from $S$.
He walks $1$ mile due east, in the process circumnavigating $S$ a total of $n$ times, arriving back at $P$.
He then walks due north again, to the place he started from.
Let us treat the location around $S$ as a plane surface.
Thus we have that:
\(\ds 2 \pi \paren {x - 1}\) | \(=\) | \(\ds \dfrac 1 n\) | Perimeter of Circle | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \dfrac 1 {2 \pi n} + 1\) |
where $n$ is a (strictly) positive integer.
$\blacksquare$
Sources
- 1965: Martin Gardner: Mathematical Puzzles and Diversions ... (previous): Nine Problems: $1$
- 1988: Martin Gardner: Hexaflexagons and Other Mathematical Diversions ... (previous): Nine Problems: $1$