White and Black play the following game. In the corners of the chessboard stand two kings: White on a1, Black on h8. The players take turns making moves. White starts. A player can place their king on any adjacent square (provided it is free), adhering to the following rules: the distance between the kings cannot be increased (the distance between two squares is the minimum number of steps the king can take to move from one square to another: thus, at the beginning of the game, the distance between the kings is 7 moves). The player who places their king on the opposite edge of the board wins (the White king on the h-file or the 8th rank, the Black king on the a-file or the 1st rank). Who will win with correct play?
Let's describe the winning strategy for white. First, the white king should move to the right along the 1st rank until the black side reduces the distance between the kings. If the black side does not do this, the distance between the kings remains 7, and the white king will reach the right edge of the board without interference.

Suppose the black side reduces the distance, then the black king will move off the 8th rank. In response, the white king also moves off the 1st rank, making a move to a square of the same color as the square the black king is on (if there are two such squares, he chooses the right one).

We will consider the chessboard as the center of a sufficiently large grid square. When both kings are on squares of the same color, we call the zone the smallest square of cells that includes both kings and in which they are centrally symmetric (the zone may partially extend beyond the board). Before the black king moves off the 8th rank, the zone appears after his moves and coincides with the board. After the white king moves off the 1st rank, a 6x6 zone appears, which does not contain cells of the edges of the board that the black king is aiming for. We will show that white can act in such a way that after each of their moves, the zone exists (for this, the white king moves to a square of the same color as the black one) and has the same property. Before the black move, the kings stand in opposite corners of the zone. The black king cannot move beyond the zone, a move along the edge does not reduce the distance between the kings, and any other move reduces this distance. A move along the edge can be a move forward or backward, that is, bringing the black king closer to the desired edge of the board or moving him away from it.

Let's consider four cases of black moves. Before this, the kings always stand in opposite cells of an even-sized zone.

1) The black side does not reduce the distance, moving forward. Then the white side moves centrally symmetrically in the zone, reducing by 1 the sum of the distances to the right and top edges of the board. In this case, the zone does not change.
2) The black side does not reduce the distance, moving backward. Then the white side moves in the same direction (for them, forward). In this case, the zone will shift by 1 in the same direction, and the sum of the distances from the white king to the winning edges will decrease.
3) The black side reduces the distance, moving off the edge of the zone. Then the white side moves centrally symmetrically in the zone, reducing the size of the zone by 2. In this case, the center of the zone does not change.
4) The black side reduces the distance without leaving the edge of the zone. This is possible only if the black king moves diagonally from one edge of the zone to another (between cells adjacent to the corner of the zone, see the figure). In this case, a zone of a smaller size will appear, and the kings will be in opposite corners of this zone, that is, on the same diagonal. Since the size of the zone has become odd, there is at least one more cell on the diagonal. The white side moves along this diagonal, reducing the size of the zone by another 1. The new zone is a part of the one that was before the black move.

![](https://cdn.mathpix.com/cropped/2024_05_06_a3c8ec1189b8a5040576g-25.jpg?height=723&width=738&top_left_y=542&top_left_x=663)

It is clear that, playing according to this strategy, white will not lose, as the black king cannot move beyond the zone and, therefore, will not reach the left or bottom edge. We will show that white will win. The black king can make only a finite number of consecutive moves of type 1: no more than 10 moves along the two edges of an unchanging zone. The zone can only decrease a finite number of times. Therefore, the black side will have to make moves of type 2 from time to time. Each of these moves brings the white king closer to the winning edges, and sooner or later he will reach one of them.

## Answer

White.

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