Help retractors

In retractor problems, one or both players take back a specified number of moves, with the goal of reaching a position where a forward task, such as a mate-in-1, can be realised. In help retractors, the two sides cooperate during the backward play, while in defensive retractors, they act in opposition. Here we shall look at the former (the more accessible type), specifically those in which both players retract one move and aim for a helpmate-in-1. Retractions in problems give rise to many interesting effects, like uncapturing an opponent’s piece and unpromoting a unit that’s on the eighth rank. Such unconventional forms of play are, nevertheless, still bound by legality – i.e. it is forbidden to take back any move that would result in an illegal position.

The best known of all help retractors is one by Julio Sunyer that contains only the two kings in the diagram position. Since that composition is examined on this site as No.132 in the Problem World article on Classics, I won’t repeat it here. Another excellent retractor appeared as Weekly Problem No.684, which features unpromotions prominently. Because of that quoted example, I will skip similarly themed problems in the selections below.

When an uncapture occurs in retractors, that effectively adds a unit on the board, and in most positions, the retracting piece has a choice of five different kinds of units to leave behind (besides not uncapturing at all). This built-in complexity to the play is perhaps why such problems tend to be short and commonly involve one solution or phase only. We start with an unusual example that manages to incorporate two nicely balanced solutions.

After White’s retraction, Black clearly needs to uncapture the eventual mating piece, but using the king on b8 for this purpose doesn’t work (e.g. 1…Ka7xQb8 and no helpmate-in-1 can be arranged with 1.Ka6/Kb6). So White wants to uncapture a black unit on either g5 or a1, to allow that unit to uncapture a potential mating piece on the same square. A queen mating move is feasible from either square only if Black could block the flight on a7, hence each uncaptured black piece has to serve this extra function. Retract 1.hxBg5 Be3xQg5, and play 1.Ba7 Qd8; retract 1.Kb1xRa1 Ra7xQa1+, and play 1.Ka8 Qh8. White’s retracting unit in each part has to choose a square that avoids interfering with the subsequent play (1.fxBg5?, 1.Kb2xRa1?). Also, if a black queen (or rook) is uncaptured on g5, it could also self-block a7, but the placement of the white king on the a-file means that such a piece would check and spoil the helpmate.

As mentioned previously, retraction moves that lead to an illegal position are prohibited; the reasons for the illegality, though, can be simple or subtle. An example of the former is that White can never retract to a position in which Black is in check, as that would imply White has played a forward move while already checking Black. Thus in the diagram above, White cannot start by retracting 1.Rf1-d1?? This composition also demonstrates a less obvious type of illegality, which necessitates “proper” retro-analysis to uncover.

Here a back-rank mate may be expected, but we run into difficulties if the black king heads for e8 (f7-flight can’t be covered) or d8 (black queen prevents mate on either side). The king can reach c8 in time if castling is utilised, and this option requires the white rook to mate from the left side. Since White’s king and rook are in the castled position, uncastling seems possible to bring this rook to the a-file quickly, and the apparent solution is: retract 1.0-0-0 Ke8-f7 for 1.0-0-0 Ra8. However, this sequence is a trap! Consider the position after reversing 1.0-0-0 Ke8-f7, with both kings and rooks on their home squares. It is Black to play, so White moved last with a unit that’s still on the board. Since both white pawns are on their original squares, this last move was made by the king or the rook, a fact that contradicts the legality of White’s 0-0-0, which was just retracted (see the castling convention).

To ensure that White’s uncastling move remains valid, the black king must respond by uncapturing a white unit on f7, which thus could have made the previous move. Furthermore, this unit on f7 cannot be a queen, bishop, knight, or pawn, because any of them would attack e8 or d8 and disable Black’s castling. Therefore the solution is: retract 1.0-0-0 Ke8xRf7 for 1.0-0-0 Ra8. The uncapture of the white rook is ingeniously forced in this problem that blends uncastling and castling play.

Our next retractor contains four parts with distinct solutions, each depending on where the black bishop begins. The white pawn acts as the thematic unit here, by executing a sort of Albino task in reverse. The Albino is a standard theme in which a white pawn on its home square plays each of its four possible moves in turn. The retraction analogue of this idea sees a white pawn on the fourth rank taking back four different moves, again the maximum possible. The solutions are: (a) retract 1.exRf4 Rg4xQf4 for 1.Kh4 Qh6; (b) retract 1.gxQf4 Qf3xQf4 for 1.Qg2 Qh4; (c) retract 1.f3-f4 Bd6xQb4 for 1.Bh2 Qg4; and (d) retract 1.f2-f4 Bc6xQe8 for 1.Bg2 Qh5. The main white pawn theme is marvellously achieved with incredible economy (four units only!) and homogeneous twinning (all shifts of the same piece). Supplementally, the white queen delivers four varied model mates, and the need to uncapture a black queen in (b) is a surprising touch for a helpmate.

The black king has two flights on e3 and f3, neither of which can be easily blocked or guarded by the existing pieces. We may assume that Black will block one of them with an uncaptured piece, while the other will be covered by the mating move, though a queen or rook mate along the e-file (observing e3) is ruled out by the f1-knight. Both solutions in fact involve Black self-blocking e3 while the mating piece guards f3.

In part (a), the white king is well placed to uncapture a rook, which has immediate access to e3 (a third black rook is legal since Black is missing a pawn). Such a rook on e3 would obviously stop a queen or bishop mate on g4, so Black wants to uncapture a knight that will mate on d4. The latter square is controlled by the h8-bishop, which can be cut off in one move by the d3-knight. Given this Se5 move will attack d7 and f7, the white king must avoid these squares when first uncapturing. Retract 1.Ke7xRe6 Re3xSe6+ for 1.Se5 Sxd4. Black’s rook retraction is a sort of critical move that goes over e5, such that the knight’s arrival on that square unpins the white mating piece. Since Se5 also cuts off the h8-bishop as noted, this move produces a double-interference effect, rather like a Novotny but with all black pieces.

The sophisticated strategies seen in (a) are all remarkably mirrored in part (b). The white king starting on g5 can uncapture a bishop that’s suitable for blocking e3. This bishop in turn may uncapture a queen on g5 that’s able to mate on g4. Since g4 is defended by the d4-rook, Black needs to unguard it with a knight interference on f4. This knight move will control g6 and h5, so the white king cannot begin by retracting to these squares. Retract 1.Kh6xBg5 Be3xQg5+ for 1.Sf4 Qxg4. Here the black bishop’s retraction is a critical move that crosses over f4, then the occupation of that square by the knight unpins the mating queen. Again, because the black knight also cuts off the rook, the Sf4 move demonstrates a same-coloured Novotny effect.

This retractor thus shows an orthogonal-diagonal transformation of complex line play across the two solutions. As if that were insufficient content, a rare formal idea is also accomplished: four different types of pieces are uncaptured during the play – rook, knight, bishop, and queen. Such a retraction task may be likened to the Allumwandlung theme (though the analogy isn’t exact, in that uncapturing an enemy pawn is a fifth option). An extraordinary piece of work that deserves much better than the Commendation it received!

Illegal clusters

29 Sep. 2022