Rats and cats

[BMAD]

Two are playing the game "cats and rats" on the chess-board 8x8. The first has one piece -- a rat, the second -- several pieces -- cats. All the pieces have four available moves -- up, down, left, right -- to the neighbour field, but the rat can also escape from the board if it is on the boarder of the chess-board. If they appear on the same field -- the rat is eaten. The players move in turn, but the second can move all the cats in independent directions. 

 

 

a) Let there be two cats. The rat is on the interior field. Is it possible to put the cats on such a fields on the border that they will be able to catch the rat? 

 

 

b) Let there be three cats, but the rat moves twice during the first turn. Prove that the rat can escape.

[BMAD]

Oh, it probably means "not on the border", doesn't it?

 

Yes. Choose one of the two diagonals going through the rat's starting position, and place the cats on the two border tiles on this diagonal. No matter how the rat moves, the cat's can always remain on this diagonal in a way that decreases the distance between the two cats with each passing turn. This means that the rat can neither escape the board nor stall the game indefinitely.

 

 

Each cat can restrict the movements of the rat in up to two directions.

 

If the rat is on the same diagonal as a cat, it cannot pass the cat in the direction of the diagonal, which corresponds to two non-diagonal directions. For example, if the cat is above and to the right of the rat, then if the rat moves up or right, the cat can reduce its distance to the rat on the diagonal. On the other hand, if the rat continually moves down or left, it can maintain its distance from the cat.

 

If the rat is not on the same diagonal as a cat (between the cat's diagonals), it cannot pass the cat in the direction of the cat. For example, if the rat is to the left of the cat, then the rat cannot pass the cat by moving right. However, the cat can never catch up to the rat if it continually moves left, up, or down.

 

When there are three cats, no matter how they are initially placed, the rat can always move to a square not on the same diagonal as any of the three cats (The area accessible to the rat on its first move can only be covered by five diagonals). The rat's movement is only restricted in three out of four directions. If the rat continually moves in the unrestricted direction, it will eventually escape the board.

[gavinksong]

What is the "interior field"?

[gavinksong]

Oh, it probably means "not on the border", doesn't it?

 

Yes. Choose one of the two diagonals going through the rat's starting position, and place the cats on the two border tiles on this diagonal. No matter how the rat moves, the cat's can always remain on this diagonal in a way that decreases the distance between the two cats with each passing turn. This means that the rat can neither escape the board nor stall the game indefinitely.

[BMAD]

nice. now for b.

[gavinksong]

Each cat can restrict the movements of the rat in up to two directions.

 

If the rat is on the same diagonal as a cat, it cannot pass the cat in the direction of the diagonal, which corresponds to two non-diagonal directions. For example, if the cat is above and to the right of the rat, then if the rat moves up or right, the cat can reduce its distance to the rat on the diagonal. On the other hand, if the rat continually moves down or left, it can maintain its distance from the cat.

 

If the rat is not on the same diagonal as a cat (between the cat's diagonals), it cannot pass the cat in the direction of the cat. For example, if the rat is to the left of the cat, then the rat cannot pass the cat by moving right. However, the cat can never catch up to the rat if it continually moves left, up, or down.

 

When there are three cats, no matter how they are initially placed, the rat can always move to a square not on the same diagonal as any of the three cats (The area accessible to the rat on its first move can only be covered by five diagonals). The rat's movement is only restricted in three out of four directions. If the rat continually moves in the unrestricted direction, it will eventually escape the board.