Four free rooks for checkmate

Question

You are given a chessboard with a black king at d5, and four white rooks in your hand.

You may place the rooks onto the board one at a time, always checking the king. After each rook placed, the king may make a single move according to the normal rules of chess.

How should you place the rooks so as to guarantee checkmate with the fourth one?

This puzzle is harder than it looks; apparently it was once presented to a group of grandmasters, who took multiple minutes to solve it.

_{There are several valid solutions (e.g. due to the symmetry of the board); finding a single one will be enough.}

_{Disclaimer: I didn't invent this idea, but nor can I credit anyone specific for it; it's a very old, anonymous, puzzle.}

Answer 1

First rook is placed at any of

A5, G5, H5, D8, D2, or D1

Second rook is placed

Two squares from the king, in the direction it moves towards, vertically if the first rook was placed on the fifth rank, and horizontally if the first rook was placed on the D file.

The third rook is placed

Protecting either of the placed rooks, preferentially one that is threatened, and checking the king on the same axis as the first-placed rook.

And then the fourth rook

On the remaining open rank or file between the first and second-placed rooks, protecting a threatened rook if necessary, yielding checkmate.

A crappy example game below:

Answer 2

First rook:

H5. It's placed as far away as possible so that it won't ever be under threat from the King until the final rook can protect it.

Second rook:

Assuming the King moves south, then X2, where X is the same column as the King (C, D, or E).

Third rook:

X3, where X is the same column as the second rook.

Final rook:

D4

The third and fourth rook placement order is:

Dependent on the movement of the King. If the King stays in row 4 after the second rook, place H4 first, then follow the instructions for the third rook above.

The solution is:

Mirrored across row 5. If the King moves north, the second rook is X8, the third rook is X7, and the final rook is H6, where X is in the same column as the King after its first move. The solution can also be reflected diagonally across the the line y = -x, where the coordinate (0,0) is the center of the board. This guarantees checkmate exactly after the fourth rook is placed.

It's been pointed out that the trivial solution for travelling north:

Is to simply place the rooks in a line.

Answer 3

First Rook

E1. (This also works if you start at H4 and "rotate" all moves accordingly)

Second Rook

If the king stays in the D rank, place the second rook at C1. If the king moves to the C rank, place the second rook at B1

Third Rook

If the second rook is at C1, checkmate happens by placing the third rook at D1. If the second rook is at B1, place the third rook either at C1 or D1.

Fourth Rook

If needed, place it at C1 or D1, whichever is empty. All rooks will protect each other, so there is no danger of the king killing any and escaping. The king is now in checkmate.