The lion plays a deadly game against a group of 100 zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the 100 zebras may arbitrarily pick their 100 starting positions. The the lion and the group of zebras move alternately:
- In a lion move, the lion moves from its current position to a position at most 100 meters away.
- In a zebra move, one of the 100 zebras moves from its current position to a position at most 100 meters away.
- The lion wins the game as soon as he manages to catch one of the zebras.
Will the lion always win the game after a finite number of moves? Or is there a strategy for the zebras that helps them to survive forever?
Answer
Zebras win.
Here is the strategy:
The zebras choose 100 vertical strips on the steppe, each of width 300m. The strips do not intersect. Each zebra promises to stay horizontally centered on its own individual strip.
If the lion enters a strip, then the zebra in that strip flees vertically away from the lion until the lion leaves the strip.
It works because:
The lion cannot kill a zebra on the turn it moves into a strip, because the strips are too wide. And if it is already inside a strip, then the zebra in that strip has just moved 100m away from it, so it cannot catch the zebra.
No comments:
Post a Comment