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Problem of the Week Interlude

The Doomed Lemming

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Here is a new problem out there that may well be too hard for my usual PoW postings....I [Stan] have not solved it! It is from a new book by Ravi Vakil called A Mathematical Mosaic: Patterns & Problem Soving (Brendan Kelly Publishing, 1996) that was reviewed in a recent American Mathematical Monthly. This problem is mentioned both in the review and in the book, but no solution is given. It was discovered by Kevin Purbhoo, a high school student in Toronto.

"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 some 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°. 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."

© Copyright 1998 Stan Wagon. Reproduced with permission.

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The Math Forum

2 October 1998