## New Problem: Lemming

I’m at summer camp now, and my dorm is split between math and physics. I hear a lot of cute math problems. Here’s a recent one that I like. It’s significantly more “mathy” than some of the other cute problems I’ve heard.

This is from Ravi Vakil’s A Mathematical Mosaic: Patterns and Problem Solving

On a remote Norwegian mountain top, there is a huge checkerboard, 1000 squares wide and 1000 squares long, surrounded by steep cliffs to the north, south, east, and west. Each square is marked with an arrow pointing in one of the eight compass directions, so (with the possible exception of some squares on the edges), each square has an arrow pointing to one of its eight nearest neighbors. The arrows on squares sharing an edge differ by at most 45 degrees. A lemming is placed randomly on one of the squares, and it jumps from square to square following the arrows.

Prove that the poor creature will eventually plunge from a cliff to its death.

### 4 Responses to “New Problem: Lemming”

1. Woods Says:

I was thinking about this briefly. The proof seems to stem from the fact that you can’t have any loops, because then somewhere inside of the loops you end up with arrows that would have to differ in pointing angle by more than 45 degrees, though I haven’t come up with an elegant method of proving this yet.

2. Mark Eichenlaub Says:

good start

3. Answer: Lemming « Arcsecond Says:

[…] By Mark Eichenlaub The problem asked: On a remote Norwegian mountain top, there is a huge checkerboard, 1000 squares wide and […]

4. Answer: Lemmings (part 2) « Arcsecond Says:

[…] Lemmings (part 2) By Mark Eichenlaub Here is another answer to the Lemming problem, complementing yesterday’s […]