On a plane, there are 100 points-sheep and one point-wolf. In one move, the wolf moves a distance of no more than 1, then one of the sheep moves a distance of no more than 1, then the wolf moves again, and so on. Can the wolf catch one of the sheep from any initial arrangement of the points?
Sheep can walk along parallel lines that are quite far apart from each other.

## Solution

Introduce a coordinate system on the plane. Let the $k$-th sheep have coordinates ($4k; 0$), $k=1,2, \ldots, 100$, and the initial position of the wolf is such that it cannot reach any sheep in one move. We will describe the strategy of the sheep. The sheep will only move along the lines $x=4k$. Suppose before the sheep's move, the wolf is at the point ($x; y$) in one of the vertical strips $|x-4k|<2$. If $z$ is greater than $y$, the $k$-th sheep will move to the point ($4k; z+1$), and to the point ($4k; z-1$) if $z$ is not greater than $y$ (i.e., the $k$-th sheep moves away from the wolf by 1 along the Oy axis). It is clear that after this, the wolf cannot catch any sheep in one move. If the wolf is outside the strips $|x-4k|<2$ before the sheep's move, the sheep can make any move. By following this strategy, the sheep can ensure that none of them will be caught.

## Answer

no.