Arrow Wide
On a remote Norwegian mountain top, there is a huge checkerboard, 1000 squares wide and 1000 squares long...?
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 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. (credit: Kevin Purbhoo, from my high school)
If you have never been on a mountain, your question is only a hypothesis, if you've been on a mountain but never a Norwegian mountain, then you're fantasizing.
I doubt you could put that large a checkerboard on a Norwegian mountaintop: 1000 x 1000 = 1000000 square ____________ [meters, miles, inches, kilometers, etc... ], the lemming will plunge to its death within minutes or seconds depending on the temperature, speed of the wind and the windchill factor!!! It doesn't matter how many degrees or angles or whatever: If the wind blows, you'd be blown off course, no matter what, especially on a Norwegian mountaintop where it's very cold most times of the year. If the windchill sets in, you'd FREEZE to DEATH especially if the lemming isn't wearing any clothing, which most animals don't, even while standing on the checkerboard, therefore, sparing the lemming from falling to a gruesome death.
But on the other hand, if this is an extraordinary lemming, they might be able to adapt to the cold weather and not Freeze to death. After playing such a boring game for many years, the poor creature with no friends might consider committing suicide by jumping off the cliff / mountain?!
What do you mean by "neighbors"? Countries? Norway is only bordered by Sweden, so it doesn't really have any "neighbors" except Sweden or maybe even Finland. You cannot call a country a neighbor if it's really not geographically connected to your own.
It'll be on at least two squares a day x number of times moved x 8 compass directions = 16x, where x = number of times moved, 1000000 / 16 = 62500 = the number of possible times the lemming could move to go though all 1 million places on the board, but it could be repeated many times until the lemming decides to commit suicide. Either freeze to death or commit suicide, it will eventually die, might not fall off a cliff but will eventually die, possibly of old age and boredom if not anything else.
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