The black king initially has two flight squares on b5 and b6, and the eventual rook or bishop mate in each solution is likely to cover one of these squares.  Black has to take care of the remaining flight by blocking it, with a piece that does not interfere with the mate.  This is possible only by means of a self-pin:

1.Rxe6 Rg6 (White shuts off the black bishop) 2.Rb6 d4.

The second solution is perfectly analogous to the first: 1.Bxd3 Be2 (cutting off the black rook) 2.Bb5 e7.

This problem demonstrates an orthogonal-diagonal transformation.  That is, the strategic effects that take place along orthogonal lines in one solution recur on diagonal lines in another solution, and vice versa.

25. David Shire
Phénix 1993
3rd Commendation

Helpmate in 2
2 solutions

26. John Niemann
Schachmatt 1947
Special Prize

Helpmate in 2
Set play

Problem 26 requires a set play to be found, in addition to the actual phase.

Set play in helpmates refers to a move sequence that solves the problem, but which is commenced by the player who normally moves second.  So in a two-move helpmate, this extra sequence goes like this: 1…White begins, 2.Black moves, White mates.  The set line is always shorter than the actual play, typically by a single move.

In the diagram position, the black king has a lot of freedom, which suggests the piece may step into a more confined area during the course of play.  White facilitates this plan by sacrificing a piece in the set line: 1…Qxd4+ 2.Kxd4 Bc3.  In the actual play, Black has no waiting move that would allow this set line to work again.  Instead, the black rook captures the other white piece so as to vacate d5 for the king: 1.Rxa5 Qc8 2.Kd5 Qf5.  The paradoxical capture of white forces is a motif frequently seen in helpmates.

Problem 27 consists of four parts – a “quadruplet” – where each solving position is created by a unified method: substituting a piece in the diagram with a new one.

The initial setting, part (a), in which a white rook stands on h7, is solved by 1.Ke2 Re7+ 2.Kd1 Re1.

For part (b), replace the rook with a white bishop, and the solution is now 1.Kg2 Bf2 2.Kh1 Be4.

Part (c) begins with a white knight on h7, changing the play to 1.Kg4 Kg7 2.Kh5 Nf6.

Finally, for part (d) use a white pawn instead, and the solution becomes 1.Ke4 h8(Q) 2.Kd5 Qe5.

The four solutions of this helpmate together produce a geometric effect: an extended star pattern formed by the black king’s moves.

27. Vojko Bartolović
Magyar Sakkélet 1956
1st Hon. Mention

Helpmate in 2
(b) WBh7
(c) WNh7
(d) WPh7

28. Zivko Milovanovic & Ewgenij Sorokin
Schach-Echo 1974
Special Prize

Helpmate in 2
2 solutions, Duplex

Yet another way for a problem to include more than a single phase is the duplex, a type mostly found in helpmates.  A duplex indicates that a problem’s task has to be fulfilled twice, once normally, and once again with White and Black exchanging their roles.  So in a duplex helpmate, an additional solution is called for in which White plays first and is mated by Black.

Problem 28 has four parts in total, two solutions for each of the duplex halves.  When Black begins, White mates with 1.e1(R) Bc6 2.Rh1 Rg2, and 1.Kh1 Kh3 2.e1(B) Bc6.  And when White starts, Black mates with 1.Rg4 e1(N) 2.Bh5 Nf3, and 1.Rg3 e1(Q) 2.Bh5 Qxg3.

The black pawn promotes to four different pieces, making this economical helpmate a remarkable demonstration of a theme known by its German name, Allumwandlung.

Another masterful work but in a very different style, Problem 29 features spectacular captures of white pieces as the main idea.

The first capture in each solution is motivated by the need to un-guard a square that the black king wants to access.  A second white piece gets sacrificed with the king move itself.

1.Qxc8+ Rd8+ 2.Kxc4 Qd4, and

1.Qxh8+ Bg8+ 2.Kxd4 Rc4.

Note that the mating piece in each solution becomes the captured piece in the other solution.  This reciprocal relationship between the two parts (also shown in Problem 26) is termed the Zilahi theme.

29. Fadil Abdurahmanović
Mat 1980
2nd Prize

Helpmate in 2
2 solutions

30. Nikolaj Dolginowitsch
Die Schwalbe 1993

Helpmate in 2
2 solutions

Solution to Problem 24 in the previous column:

1.Kf2! (waiting)

1…h2 2.Nb6 axb6 3.a7 b5 4.a8(Q); and

1…Kh2 2.Ne3 Kh1 3.Nf1 h2 4.Ng3.

24.  A. Moozhoor
The Problemist Supplement 1993

Mate in 4