[Event "Beginner Endgames You Must Know!: The rook box with better defense"]
[Site "https://lichess.org/study/wukLYIXj/tGmDrQtl"]
[Date "????.??.??"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "*"]
[Annotator "https://lichess.org/@/NoseKnowsAll"]
[ECO "?"]
[FEN "8/8/8/8/8/3k4/R7/3K4 w - - 0 1"]
[Opening "?"]
[SetUp "1"]
[UTCDate "2021.05.24"]
[UTCTime "16:53:15"]
[Variant "Standard"]

1. Re2 { Now let's see what happens if black plays smarter and realizes that stepping towards any of the four walls will lead to a fast checkmate.
PRO TIP: If you're looking to survive the longest in any endgame, stay central with your king. It is much easier to get checkmated against a wall than in the center of the board. [%cal Be2e8,Be2a2] } 1... Kd4 2. Kd2 { Our rook could not make a move that shrunk the box, so we take the time out to improve our king. This moves takes opposition (and therefore the important c3 and d3 squares from our opponent), shrinking the box a tiny bit. } 2... Kd5 3. Kd3 { If our opponent lets us, we can continue to take opposition - black has to keep giving ground. } ( 3. Re3 $2 { Reasonable, but this move doesn't make progress. } 3... Kd4 { Now we see it's us who has to give up ground! The rook cannot move because then the king escapes the box. The king cannot move because it must protect the rook. } ) 3... Kc5 { And here's the only tricky move to remember in this checkmate. } ( 3... Kc6 4. Re5 { Mission accomplished - the box is being shrunk } 4... Kd6 5. Kd4 { And we've seen this position already. Black has to continue giving ground. } ) ( 3... Kd6 4. Kd4 { The box continues to be shrunk... } ) 4. Ke4 $3 { Because black is not threatening to escape the box created by our rook for one turn, our king is allowed to cross over to the other side of the e file in order to force the king back. [%csl Bd5][%cal Be2e8,Be2a2] } 4... Kc4 ( 4... Kd6 5. Kd4 { Taking opposition. We've seen this position above. Black has to continue giving up ground. } ) 5. Rc2+ { Opposition allows us to continue shrinking the box. [%cal Ge4d3,Ge4d4,Ge4d5,Bc2c8] } 5... Kb3 6. Kd3 Kb4 7. Rc3 Kb5 8. Rc4 { [%cal Bc4c8,Bc4a4] } 8... Kb6 { The box has been shrunk as much as we can with just the rook. Now we again involve our king to force the black king further back. } 9. Kc3 Kb5 10. Kb3 Kb6 11. Kb4 { We continue the pattern. Within the box defined by our rook, improve the king as much as possible until we hit the critical juncture here. Black has to give up ground by either going towards the side of the board or give up ground by going towards the top of the board. [%cal Bc4c8,Bc4a4] } 11... Ka6 12. Kc5 $1 { Compare this position to the position on move 4. When the king is not threatening to leave the box created by our rook, our king is able to crossover the line in order to cover the b6 (in the previous position d5) square. [%csl Bb6][%cal Bc4c8,Bc4a4] } 12... Kb7 ( 12... Ka5 $6 13. Rb4 { With the king trapped on the side of the board, we know how to chase him all the way to a corner. } ) 13. Kb5 Ka7 14. Kc6 $1 { Crossing over again. [%csl Bb7] } 14... Kb8 ( 14... Ka6 $6 15. Ra4# ) 15. Kb6 Ka8 16. Rc8# { The king is trapped. No need to chase him any further.
To remember this checkmate, ask yourself two questions: (1) Can I use my rook to shrink the box around the king further? If so, make a rook move. If not, (2) Where do I need to position my king to keep shrinking this box further? This usually means taking opposition from our opponent and push him back more. } *