Run Brilli, Run!
Brilli the ant is trapped in a grid, along with anteaters!
All the anteaters, and Brilli herself, can only stay on the vertices of the unit squares. In each move, Brilli and the anteaters can,
- either stay in the same vertex,
- move to a vertex which is at a distance of unit from the current vertex.
In each move, Brilli moves first, and the anteaters move after Brilli.
Brilli faces a tragic end if an anteater comes to the same vertex as her. Brilli does her best to survive.
Given that Brilli and all the anteaters are Brilliant and try to reach their goals, what is the minimum value of for which the anteaters definitely win?
The answer is 2.
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Clearly, 1 anteater doesn't work. To prove this, consider 3 cases in which the anteater could catch Brilli in the next two moves without Brilli moving towards it without being forced to:
Case 1, Brilli is on a corner, and the anteater diagonal from Brilli. In this case, Brilli can stay still, and if the anteater also stays still, we are back to this same case. If the anteater moves, Brilli can escape.
Case 2, Brilli is on a corner, and the anteater is on the same square, but not diagonal from Brilli. Thus, Brilli can escape.
Case 3, The locations of Brilli and the anteater are none of the above. In this case Brilli and the anteater are the same position relative to each other, but not next to an edge. In this case, Brilli has more escape options than above, in adition to being able to escape in the same ways as above.
To prove that 2 anteaters can catch Brilli, the grid will be notated as if on the cartesian plane, with vertices at (0,0), (0,2014), (2014,2014), (2014,0).
To start, one of the anteaters will try to get on the same x coordinate as Brilli.
This can be done by simply moving towards the x coordinate or Brilli. Either Brilli moved away, or the anteater gets at least 1 step closer. Brilli cannot move away indefinitely, as she will eventually hit the edge of the grid, and thus the a single anteater can eventually get on the same x coordinate as Brilli.
When Brilli moves right or left, this anteater will do the same. When Brilli doesn't change her x coordinate, the anteater will just move 1 step closer to Brilli. In this way, Brilli can never pass the Y coordinate of this anteater, and must continue to move left or right, or else this anteater will further restrict the Y axis of where Brilli can be. The other anteater creates a similar restriction with being on the same Y coordinate as Brilli, and thus each move she must let 1 anteater get 1 unit closer to her, or move 1 unit closer to an edge, which she cannot do indefinitely. Eventually, the distance between Brilli and one of the anteaters will eventually reach 0, and Brilli will be dead.