Carving problems with shortest games

Most of our quoted examples come from Alain Brobecker, a French problemist and retro expert. He’s a programmer as well, and apparently he wrote software to generate these carving problems. When solving this retro type, I tend to first look for potential mating moves while assuming most units standing on a game-array square are the original occupants. Naturally, we should also be mindful that such apparent home-units could have been replaced, possibly even by a piece of the opposite colour. Once a plausible mating scheme is found, we check that the position is attainable in the given number of moves.

In the first diagram, the absent f7-pawn hints at a kind of “fool’s mate” on the light diagonal, but if we aim for Qh5 mate, the impeding g7-pawn can’t be replaced within the three moves available to reach the position. Alternatively, White may use the seemingly missing f1-bishop to deliver the mate, and this leads to the idea that the unit standing on h7 is that piece, ready to mate on g6. For the stipulated game, the bishop can easily reach h7 in two moves after Pe3, but there’s no time left to capture the f7-pawn. Therefore that black pawn is still on the board, masquerading as a white one. 1.e3 f5 2.Bd3 f4 3.Bxh7 fxe3 for Bg6 mate.

In the Chess960 variant, the pieces on the 1st and 8th ranks are randomly placed at the start of a game, with the opposing ones on the same file matching in type. Two additional rules for the initial set-up are: (1) the bishops are added on different coloured squares, and (2) the rooks are placed on opposite sides of the king. Given the wealth of possible starting positions in Chess960, it’s pleasing how we can narrow them down to precisely one in the next two carving problems, when the usual pair of conditions is applied.

Since the position is reached after just two moves, White most likely shifted a knight from g1 to e5, where it could serve as the mating piece. The king never starts in a corner in Chess960, because the rooks are required to be on its either side; so Sxf7 doesn’t work as a mating move against a king on h8. The only possible knight mate is on d7, meaning the king is on f8. The quickest way to account for the empty c8-square is to have a knight start from there, and while the piece couldn’t have reached e5 in time to be captured, it could have replaced the e7-pawn which made the sacrifice. Hence the game is 1.Sf3 e5 2.Sxe5 Se7 for Sxd7 mate. By white-black symmetry, the other black knight began on g8; then the only right-side square left for a rook is h8. To avoid a guard on the d7-mating square, e8 must be occupied by the other rook, and d8 by the dark-squared bishop. The light-squared bishop and queen are thus on the remaining squares, a8 and b8 respectively.

The next Chess960 problem is far trickier, despite not involving a single capture. Given the two-move limit, White’s play is clear: Pg3 and Q(g1)-g2 – not R(g1)-g2 since even if a queen/bishop were placed on h1, no battery mate on the long diagonal can be arranged. With the queen on g2, the mating move is likely to be Qxb7, where the piece is supported by a bishop on h1. The natural spot for a mated black king is c8 (not the a8-corner as discussed), and then we only need to account for the a5-unit. Using a knight from b8 with …Sc6-a5 fails, however, because the piece would prevent the queen mate. An alternative option is to start with a bishop on b8, which enables the play …Pa5 and …Ba7. But this leads to an illegal set-up, because the mating scheme requires a white bishop on h1, which implies – by symmetry – that Black’s dark-squared bishop had begun on h8, not b8.

There is one last possibility, namely that the king is not on c8 after all, but a7, having moved from b8! The game is thus 1.g3 a5 2.Qg2 Ka7, and White can still mate by Qxb7. We determine the rest of the black pieces as follows. The king starting on b8 means that a8 is the only left-side square for one of the rooks. The light-squared bishop can’t be on c8 where it would stop the mate, nor g8 (occupied by the queen), so it must be on e8. Only three squares remain, and the knights must avoid d8 as either would guard b7, so they go on c8 and f8, and that leaves d8 for the other rook.

We return to orthodox chess for our last selection, a difficult one which entails noticeably longer games (six moves) for the two parts. For part (a), the game must reach a position where White has a single mate-in-1, and for (b), the game leads to a different position where White has two possible mating moves. Note that in the previous examples, although the shortest game solutions involve a precise sequence of moves, that’s not a requirement for a sound carving problem. So long as the final position is unique in satisfying the conditions, the game to reach it need not be exact.

Here the total lack of captures implies that all pawns are still on their original files – the e3-unit is definitely a white pawn, for instance. An obvious candidate for the h3-unit is the bishop from f1, though it could also be another piece (white or black) that got replaced on its home square, e.g. it could be a white pawn if something else stands on h2. If Black had played …Pd5, the king might be on d7, d8, or c8 now, but then arranging a mate-in-1 proves unfeasible. For example, the king on d7 could allow a queen mate on d5, but there’s no possible departure square for the queen, which can’t replace, say, the pawn on g2 without creating an illegal position.

So let’s suppose that the king stays on e8, then a white bishop on h3 could be used to mate on d7, except that the latter square seems well-protected by Black’s c8-bishop, d8-queen, and king. However, if we envisage a white queen standing on c8 instead of the bishop, such a mate becomes viable. Not only does the queen provide a guard on d7, the piece also pins its black counterpart on d8. That means the d1-unit has to be an impostor, and only the black bishop (not the knight from b8) is fast enough to play this role. Part (a) is hence solved by 1.e3 d6 2.Bd3 Bh3 3.Qg4 d5 4.Qc8 Bg4 5.Bf5 Bd1 6.Bh3 Sd7 for Bxd7 mate. (Alternative routes are possible, e.g. 1.e3 d5 2.Qg4 Bd7 3.Be2 Bb5 4.Qc8 Sd7 5.Bg4 Be2 6.Bh3 Bd1.)

We might assume that part (b) is thematically related and the final position won’t be wildly dissimilar. Two mating moves need to be prepared, and the scheme of part (a) nearly works here in that Q(c8)xd7 is also a potential mate. To make this effective, the black pieces need to be slightly reorganised, such that pinning the d8-queen becomes unnecessary, and this is done by switching that queen and the knight on d7. However, this set-up costs extra time as it involves a 3-move manoeuvre (…Qd7, …Sc6-d8), which forces the black bishop to take the shortest route of …Bg4-d1 to compensate. These bishop moves in turn clash with the white queen’s route of Qg4-c8, with the result that the game would exceed the 6-move limit. The only way to resolve these issues is to switch the placements of the white queen and bishop as well! A sample game is thus 1.e3 d5 2.Qh5 Bg4 3.Bd3 Bd1 4.Bf5 Sc6 5.Bc8 Qd7 6.Qh3 Sd8 for Bxd7 or Qxd7 mate. 

We see an excellent doubling of the exchange of identities theme, with both pairs of WQ/WB and BQ/BS swapped across the two parts. Various impostor ideas, including exchange of identities in two phases, are known in standard proof games, but they necessarily involve two pieces of the same type and colour, in order to be deceptive. Carving problems, by enabling unlike pieces to serve as impostors, take these themes to striking new territories.