A “Holy Grail” of proof games attained

In a so-called “massacre” proof game, just a handful of pieces remain in the diagram position, and the preceding play largely consists of captures by both sides. The nature of such games, where virtually no clues of what transpired are left in the final position, means they require computer assistance to be constructed (or solved). There’s a sort of “Holy Grail” for problemists to find a massacre game with a unique solution that leaves only the two kings on the board. Earlier research by programmers had proved that no such games exist in 16½ or 17 moves (except those with additional problem conditions). However, by extending the length of the games to be analysed and limiting the search based on how far the kings will travel, Francois Labelle of Canada has discovered the first two-king PG that is completely orthodox.

This astounding achievement took a staggering amount of computer time. As Labelle reports in an article published in StrateGems last year, ten months were spent on the searching parameters that yielded the successful game in 19½ moves. The problem position, shown above, is quite nice to boot – a symmetrical set-up with both kings on their original file. The solution is 1.c4 e5 2.Qb3 Qh4 3.Qxb7 Qxh2 4.Qxb8 Qxg1 5.Rxh7 Rxb8 6.Rxg7 Rxb2 7.Rxf7 Rxa2 8.Rxd7 Rxd2 9.Rxa7 Kxd7 10.Rxc7+ Kd6 11.Rxc8 Qxg2 12.Rxf8 Kc5 13.Rxg8 Rxg8 14.Bxg2 Rxg2 15.Sc3 Rxf2 16.Kxf2 Kxc4 17.Kf3 Kxc3 18.Bxd2+ Kxd2 19.Ke4 Kxe2 20.Kxe5.

Labelle’s method also generated 349 sound PGs with three pieces left, some confirming previously known results, but plenty of which are new works. Here’s an interesting one that shows the black king trekking to the farthest possible square, in contrast to the stationary white one. 1.d4 h6 2.Bxh6 c5 3.Bxg7 Rxh2 4.Bxf8 Rxh1 5.Bxe7 Kxe7 6.Sc3 Kd6 7.dxc5+ Kxc5 8.Qxd7 Rxg1 9.Qxb7 Rxg2 10.Qxa7+ Kb4 11.Qxf7 Rxf2 12.Qxg8 Rxe2+ 13.Bxe2 Rxa2 14.Bd3 Qxd3 15.Qxc8 Rxb2 16.Qxb8+ Kxc3 17.Qxb2+ Kxb2 18.cxd3 Kxa1. While it doesn’t sound very likely, I wonder could any of these three-unit problems be “twinned” with one another!?

Are there more two-king PGs to be discovered? It’s possible, according to Labelle, whose analysis isn’t exhaustive (though he has conducted a full search of 18-move games and determined that the task cannot be accomplished at that length). Out of more than 3500 possible two-king positions, he has verified the PGs for about half, from which one sound setting emerged. Based on this and the data from the three-unit problems, Labelle estimates the chance of another two-king position having a unique solution to be about 1/1000. “Finding it,” he writes, “will not be easy and I fear it will require more computing power, a better pruning method, or plain luck.”