K+R vs. K / Infinite Board With One Corner

Here is a chess problem that people swear is impossible. On an infinite board, with one corner (see the first diagram), checkmate the lone King with King and Rook. The pieces can be placed at random, finite distances from the corner. The task is toughest when the White King is in the corner. The diagrammed position is relatively short, so that you can see how it is done. But it can be done with the Black King way out there. This article would be much longer, then.

I begin: 1 Rd1 Kc2 2 Rd9 (White need not be this efficient) 2...Kc3 (2...Kc1 3 Rd3 is similar, and takes about a move longer) 3 Kb1 Kc4 4 Kc2 Kc5 5 Kd3 Kc6 6 Ke4 Kc7 7 Kf5 Kc8 8 Rg9 Kd8 9 Kg6 Ke8 10 Kh7 Kf8 11 Kh8 [2nd diagram]

This completes the first part of White's task. And that was the part that seemed impossible, at first. To get here, White's King moved almost directly to h8. Black's King, on the other hand, took a longer path, as he was blocked by the Rook. You can probably see how to do this when the Black King is farther from the corner.

The rest is simple: 11...Ke8 12 Kg7 Ke7 (A little study shows that nothing is much better) 13 Re9+ Kd8 14 Kf8 Kd7 15 Rd9+ Kc8 16 Ke8 Kb8 17 Kd7 Kb7 18 Rb9+ Ka8 19 Kc8 Ka7 20 Rb10 Ka6 21 Kc7 Ka5 22 Kc6 Ka4 23 Kc5 Ka3 24 Kc4 Ka2 25 Kc3 Ka1 26 Kc2 Ka2 27 Ra10++ (Perhaps Black could have lengthened the solution a move or two).

That was fairly short.