White to play and win - only moving the rook once

Question

[Title "White to play and win - only moving the rook once"]
[fen "k7/8/8/8/8/8/8/K1R5 w - - 0 1"]

This is an exercise in using the opposition and outflanking from John Hawkins' Amateur to IM. Obviously your engine isn't going to help.

Answer 1

This is a simple one. You can't make progress once the K's are opposed and thus the only solution is outflanking.

[FEN "k7/8/8/8/8/8/8/K1R5 w - - 0 1"]

1. Ka2 Ka7 {is the human approach} (1... Kb8 {forces thinking about opposition} 2.Kb2 {maintaining opposition} Ka8 {since moving up sustains the white opposition, but makes it shorter} (2... Ka7 3.Ka3 {opposition but shorter}) (2... Kb7 3.Kb3 {opposition but shorter}) 3.Kc3 {outflanking = not losing opposition} Kb8 (3...Ka7 4.Kc4 {more of the same, doesn't shorten Blacks imminent death}) (3...Kb7 4.Kb3 {opposition again}) 4.Kb4 {and the opposition story goes on...}) 2. Ka3 Ka6 3. Ka4 Kb6 4. Kb4 Ka6 5. Kc5 {Ka5 won't lead to a solution.} Kb7 6. Kb5 Ka7 7. Kc6 Ka8 8. Kc7 (8. Kb6 {is a bit slower and as GM Ben Finegold would thus say: "wrong!"} Kb8 9. Ka6 Ka8 10. Rc8#) 8... Ka7 9. Ra1#

EDIT: added some more sidelines on request. IMO 4 deep for a sideline is more than sufficient to keep it readable.

Answer 2

Since it can't be analyzed with an engine, I made it so :-) I hope you agree that this setting keeps the relevant restriction (the wR only moves once):

[FEN "8/p7/rp6/rp6/pP6/R7/8/K6k w - - 0 1"]

Note that Kb3 is impossible here, so you must be a bit careful with outflanking. But Kb1 is a clear #11. (A problem solving engine could give all shortest variants, but that takes tiiiime...)

Answer 3

Answer is edited. Initial conclusion was that it's impossible.

First of all, the only rook's move must be the last move (you cannot check by a king), so it's all about kings dancing. White are willing to checkmate here either when the black king is on a column opposed by the white king on c column or when the black king is cornered at a8 with the white king on a6/b6/c7. First option is typically avoidable by black so white's strategy is to gradually force the black king into the corner by using opposition.

Position that forces a king to step back is when it's opposed by the other king and the distance is exactly one row. For the king whose turn is next it's bad opposition, for the king just moved it's good opposition. In this play whenever black happen to obtain good opposition they can maintain it by mirroring the white king's moves (rook will not interfere here as it will always be blocked by the king). This way black keep odd number of rows between the kings. It works vice versa - whenever black ends a turn opposing the white king with odd number of rows between them - they are safe. So, good and bad oppositions should be extended from single row to any odd number of rows. Initially white can immediately obtain good opposition by 1. Ka2 but the catch here is that white need to approach the black king and black do not have to.

So the detailed strategy for white should be this:

1. Always avoid black's good opposition
2. Do not let black king to the rook
3. Approach the black king
4. Use threat of opposing the black king on a column from c

Strategy for black should be this:

1. Obtain good opposition whenever possible
2. Threaten to get to the rook (when white king is to the right of c column)
3. Preferrably maintain odd number of rows to the black king
4. Keep distance from the white king

[FEN "k7/8/8/8/8/8/8/K1R5 w - - 0 1"]

1. Ka2 {b1 or b2 allow good opposition for black} Kb8
    (1. Kb1 Kb7!)
    (1. Kb2 Kb8!)
    (1... Ka7 {allows good opposition for white} 2. Ka3)
    (1... Kb7 {allows good opposition for white} 2. Kb3)
2. Kb2 Ka8
    (2. Ka1 {same for 2. Ka3, allows good opposition for black} Ka7! {black are safe})
    (2. Kb1 {same for 2. Kb3, allows good opposition for black} Kb7! {black are safe})
    (2... Ka7 3. Ka3)
    (2... Kb7 3. Kb3)
3. Kc3 Kb7
    (3. Ka2 Kb8 {Repeats the first turn})
    (3. Kc2 Kb8. 4. Kc3 Kc7! {black are safe}
        (4. Kb2 Ka8 {Repeats the second turn})
        (4. Kd2 {Or any other to d column} Kb7 {black king gets space and approaches the rook})
        (4. Kb1 {same for 4. Kb3} Kb7!)
    )
    (3... Ka7 Kc4 {white is just getting closer})
    (3... Kb8 4. Kb4)
4. Kb3 {The white king is opposing similar to 1 ...Kb8 2. Kb2 but it's two rows closer. Now white can repeat this 2-move pattern to approach the black king} Ka7 (same reason as for 2... Ka8)
5. Kc4 Kb6
6. Kb4 Ka6
7. Kc5 {Now he threat Ra1# is active} Kb7
    (7... Ka5 8. Ra1#)
    (7... Ka7 8. Kc6 {repeating 7. Kc5 but one row closer to the edge})
8. Kb5 {same as 6. Kb4 but closer to the edge} Ka7
9. Kc6 Kb8
10. Kb6 Ka8
11. Rc8#

And remarkably, it matches the 11 turns produced by engine for edited position. For me, it's not "simple one" ©.