Math problems

In no particular order, and the headings are possibly misleading.

Graph theory

  1. Imagine the equator runs through an island, and two people are standing on opposite ends of the island where the equator meets the shore. Each has a satellite phone which displays the location and elevation of both people and allows them to communicate. Assuming the interior of the island stays above sea level, is it possible for them to hike along the equator and meet on the island all while maintaining the same elevation as each other? (see Exer. 1.3.10 of West's "Introduction to Graph Theory," 1st Ed.)
  2. "An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?" (from a Hungarian math competition, via Ted Alper)
  3. Does \(K_{1,2,2,2}\) have a finite planar cover? (This would settle Negami's Conjecture.)

Algebra

  1. Why is downtown always in the center of a city? (communicated by either Sam Roven or David Clancy)

Geometry

  1. Given four points \(a,b,c,d\) on a line in the plane, construct a square such that the parallel lines through one pair of the square's opposite sides intersect \(a\) and \(b\)... and the parallel lines throught the other pair of the square's opposite sides intersect \(c\) and \(d\). (from a Hungarian problem book)
  2. If \(\alpha\) is an irrational number, can you tile a \(1 \times \alpha\) rectangle with finitely many squares? (from Matoušek's "Thirty-three Miniatures")

Number theory

  1. Let \(f_n\) denote the \(n^\text{th}\) Fibonacci number with \(f_0 = 0\) and \(f_1 = 1\). Prove that, for any set \(A\) of non-negative integers, \(\gcd \left ( \{f_a : a \in A \} \right) = f_{\gcd(A)}\). (communicated by Dean Hoffman)

Topology

  1. Imagine a rope is pinned to a wall at its middle. Person 1 ties a knot in the left side of the rope, and then pins the left end of the rope to the wall. Person 2 ties a knot in the right side of the rope, and then pins the right end of the rope to the wall. Now, the pin in the middle of the rope is removed so that the rope hangs only by its ends. Is it possible that the rope can be maneuvered so as to untie both knots? (seen in a video of John Conway)
  2. Imagine you have a photo frame which you want to hang on a pair of side-by-side nails sticking out of the wall. But you have a bizarre stipulation for your installation: you want the picture to hang when both nails are in the wall, but you want the picture to fall if either nail is removed. Is this possible? (communicated by Lucas Van Meter)
  3. Prove that the coefficients of the Conway polynomial of the \((m,n)\)-Turk's head link merely alternate in sign compared with those of the \((n,m)\)-Turk's head link. (This is a conjecture by A. Takemura.)