In the comments onmy post about φ, Polymath, (whose [blog][polymath] is well worth checking out) provided a really spectacular [link involving φ][desert]. It's an excerpt from a book called "[Mathematical Gems 2][gems]", showing a problem that came from John Conway, called the "Sending Scouts into the Desert" problem.
The problem is:
Suppose you're given a checkerboard with all of the squares on the bottom filled. You're allowed to do standard checks jumps (jump over a man and remove it), but you can't jump diagonally, only up, left, or right. How far *up* can you get a man? How many men do you need to move to get that far?
To get to the first row above your men, you need to jump one man. To get two rows, you need to jump four men; three rows, you'll need 8 men; four rows, you'll need 20 men. How about five rows?
You *can't* get five rows doing up and sideways jumps. Even if you had an infinitely wide checkerboard, with as many full rows of mens as you want.
The proof, along with a nice Java applet to let you try the solvable ones, is at the link. And φ is an inextricable part of the proof!
[polymath]: http://polymathematics.typepad.com/
[desert]: http://www.cut-the-knot.org/proofs/checker.shtml
[gems]: http://www.amazon.com/gp/redirect.html?link_code=ur2&tag=goodmathbadma-…
Does anyone know an easy way to complete the proof? It seems intuitive that if you can't have even a single empty space, it can't be done at all, but it'd be nice to say "and because Q, even with infinite rows of infinite columns of checkers, you can't do it".