In Problem 31, Black’s king is already confined completely, but the queen prevents the white knight from mating on c1 or c3.  If the black rook were to replace the queen as the blocking piece on a1, the knight mate on c3 would work.

However, Black must take care in relocating the queen, since after 1.Rf1 Sg3, for example, any queen move will either hinder Ra1 or keep c3 guarded.

The solution is to hide the queen on g1: 1.Qg1 Sg3 2.Rf1 Se4 3.Ra1 Sc3.  The queen makes a clearance move on the rank, to allow the rook to travel on the same line and in the opposite direction – this is the Turton manoeuvre.

The convention that Black commences play in a helpmate is sometimes broken, as there is no particular reason why White should not make the first move in this genre.  Problem 32 illustrates this less common form – its stipulation of ‘Helpmate in 3½’ indicates that White begins and mates on the fourth move.

Here the four moves available are not sufficient for White to both control the b6-flight and position a mating piece, so Black has to arrange a self-block on that square.  Black only has three moves to do this with a promoted piece, and if the promotion were to occur on c1, the new piece would not reach b6 in time.

But with White’s help, Black can promote on another square so as to quicken the task: 1…Ba2 2.c2 Bb1 3.cxb1(R) Se4 4.Rb6 Sc5.

Longer helpmates in which White’s main force is a rook and a bishop often culminate with them giving a double-checkmate.

Such a mate carries the advantages of covering many flight squares, and being impervious to interposing by the black pieces blocking other flights.

Problem 33 is difficult to solve, even with this double-checking scheme in mind.

White’s rook and bishop are initially pinned, and both need to be freed before they could set up a discovered attack.  Black seems to require too many moves to unpin them, e.g. 1.Rd1 2.Rc1 3.Bg5 4.Bh6.

The diagram of 35 provides few clues as to where the black king will end up.

Testing the squares in the king’s vicinity, we find a good prospect in e5: the king placed there would give a near-complete mating position, requiring only an additional block on f4.

Black’s choice for obstructing f4 is the f8-bishop.  The play of this piece, coupled with the king’s need to access e5, force the white bishop on g7 to take an interesting trip.

1.Kf4 Bf6 2.Bh6 Be7 3.Ke5 Bf8 4.Bf4 Bg7.  The white bishop executes a rundlauf, i.e. a roundabout trip by a piece that finishes on its original square.