Weekly Problems 2020-A

A good key 1.Qh4! grants a flight on e5 and sets up a battery on the fourth rank, which threatens to open with 2.Sg6. The black knight has three defences that allow the h8-bishop to cover the e5-flight, and they produce three more battery mates: 1…Sh5 2.Sxh5, 1…Se6 2.Sxe6, and 1…Sxf5 2.Sd5 – the latter a pin-mate where White regains control of e3 and d4. A switchback occurs if Black takes the offered flight: 1…Kxe5 2.Qe7. Lastly, 1…d2 is answered by 2.Re2. The three battery variations show a duel between the two knights.

Andy Sag: A flight-giving key with switchback. I like the two variations that self-pin the black knight.
Jacob Hoover: The black knight has three defenses, and each one forces a different knight discovery.
Ian Shanahan: The two star variations are 1…Kxe5 2.Qe7 (the black king takes the flight proffered by the key, met by a switchback and pin-mate), and 1…Sxf5 2.Sd5 (the Schiffmann theme). A fine problem in traditional style.

The white knight on f2 has four moves that threaten mate, but three are defeated by its capture while 1.Sg4? is neutralised by 1…Rxb6! The key 1.Ba4!, threatening 2.Be8 and 3.Bxf7, induces Black to disable the four potential defences against the knight moves. 1…c6/Rxa4 2.Sg4 and 3.Sxf6. 1…Bb1 2.Sh1 and 3.Sxg3. 1…c3 2.Sd3 and 3.Sf4. 1…d4 2.Se4 and 3.Sxf6/Sxg3. Consistent white knight play and in three variations Black deftly handles the threat by activating the a2-bishop.

Jacob Hoover: White unravels this problem with 1.Ba4!, sacrificing the bishop to the a6-rook. In each line a black unit either is decoyed away from guarding a vital square or blocks a guard of a vital square.
Andy Sag: The four set plays by the f2-knight make a nice theme.

A white bishop move along the long diagonal will open the e-file and threaten 2.Qe4, a mate that requires the queen to observe both e6 and c4. Black can defend by placing a knight on e5 or d4, moves that also pin a white rook and thereby prevent it from mating on an unguarded square. In two thematic tries the bishop is placed to interfere with one of these pin-lines: 1.Bg7? Se5 2.Rxd6 (1…Qxe3 2.Sxe3, 1…Sd2 2.Sc3) but 1…Sd4! refutes, and 1.Bb2? Sd4 2.Rxc5 (1…Qxe3 2.Sxe3, 1…Sd2 2.Sc3/Rxc5) but 1…Se5! refutes. The key 1.Ka6! creates a new threat of 2.a8=Q, against which the knight defences also work. But thanks to the anticipatory unpinning effect of the king move, these defences no longer counter the rook mates: 1…Sxe5 2.Rxd6 and 1…Sd4 2.Rxc5. The key-move controls b5 so that if Black takes the flight, the threat is still playable: 1…Kc6 2.a8=Q; hence not 1.Kc8? Kc6! Another good try is 1.Re7? (2.Qe4) Sd4 2.Qxd4 (changed mate), 1…Bxe7 2.Sb6, but 1…Qxe3!

Andy Sag: The key breaks two half-pins. It also gives a flight which is illusory as it fails to defend the threat. The bishop tries involve the same defences which use rook pins to work.
Andrew Buchanan: Found this very hard to solve, because of so many tries. I was particularly enraptured by 1.Bb2? and 1.Bg7?
George Meldrum: The tries 1.Bb2 and 1.Bg7 look tempting yet fail. Next attention is drawn to the innocuous pawn on a7 but immediate promotion fails. The flight-giving key looks an unlikely move. So many good problems have an obvious key – this is not one of them.
Ian Shanahan: The rook-mates thwarted in the tries occur as mates after their black refutations in the post-key play! A classic reversal-pattern. A heavy, but satisfying problem.

White mating replies are prepared against all possible black moves in the diagram: 1…Sc~ 2.Re6, 1…Se~ 2.Sdc6 (dual 1…Sd5 2.Sdc6/Sbc6), and 1…e3 2.Sf3. But White has no waiting move capable of preserving all of the set play, e.g. 1.Kxg3? Sxf5+!, 1.Qg5? h4! The unexpected key 1.Sb5! (waiting) abandons not only both set knight mates but also the rook mate on e6 (by unguarding f5). Three new mates arise, however, due to a multitude of effects brought about by the key: 1…Sc~ 2.Rxe4, 1…Se~ 2.Rxc5, 1…Sd5 (correction move) 2.Sc6, and 1…e3 2.Qd4. The key-piece has to choose b5 to place an extra guard on d6, otherwise 1…Se6! would be too strong. The composer Cox had a knack for creating mutates that are difficult to solve.

Andy Sag: A waiter with four variations including three changed mates. Can convert to a Meredith by shifting the white king to e1, removing the g-pawn and moving everything else down one square.
Jacob Hoover: A rather nice block-mutate.
George Meldrum: Heaps of changed mates; dual eliminated. Nice.
Ian Shanahan: A lovely all-change mutate with a black correction in the post-key play. Very pretty and elegant!

The attractive diagram position exemplifies one-row asymmetry, where a symmetrical set-up nevertheless yields a single solution, by utilising the board edge in some way. 1…Kf5 2.Sf3 g4 3.Se5 g5 4.Sf7 g6 5.Kh6 g7 6.Sh8 gxh8=Q. Good black and white interplay ensures that the knight route to h8 is unique, and there’s an ideal-mate finish.

Andy Sag: A pawn mate is possible using the knight as a self-block but that takes seven white moves, so promotion to a queen on h8 is clearly indicated. The black king must wait for the pawn to arrive at g6 before moving and the knight must go via e5 to keep the g-file clear thus ensuring unique play.
Andrew Buchanan: Crafted with Ian’s usual elegance and attention to detail. A pleasure to see why the black knight could not land on any other square on the h-file.

The B + R/R half-battery on the long diagonal and the Q + S battery on the first rank are both controlled by the black queen. If either rook moves off the diagonal, White will threaten Rf4 by the other rook, shutting off the queen. Random moves of the two rooks are mostly defeated by 1…Bf5!, which not only stops Rff4 directly but also cuts off the g6-bishop to create a flight on b1. The thematic try 1.Rb6? forestalls 1…Bf5 by guarding b1 and it leads to 1…Qd6+ 2.Rdxd6, 1…Qh3 2.Rd3, and 1…Qxg2 2.Rd2 (plus various Q + S battery variations similar to the post-key play), but 1…Qg1! refutes. The key 1.Rd7! (threat: 2.Rff4) prevents the bishop defence and produces 1…Qd6+ 2.Rfxd6, 1…Qh3 2.Rf3, 1…Qxg2 2.Rf2 (three changed mates compared with the virtual play) and 1…Qg1 2.Rf2. Further, 1…Qg3 2.Sf2, 1…Qe5 2.Se3, 1…Qc7 2.Sdc3, and 1…Qb8 2.Sdb2 show more precise battery shut-offs. Including the virtual play, we see ten distinct battery variations (counting 1…Qxg2/Qg1 2.Rf2 as one), all instigated by the black queen.

Andy Sag: The diagonal half-battery gives a strong clue and the first rook must block the c8-h3 diagonal but the correct rook must be chosen because 1.Re6? fails to 1…Qg1/Qh3! All eight mates are battery mates including the threat.
Andrew Buchanan: Black is threatening 1…Qd6+ [an unprovided check], and that forces the key to be by one rook.
George Meldrum: But which rook move? That that stops Black playing 1…Bf5 and provides a mate after Black playing 1…Qg1. Eight mates in a cornered king problem; what’s not to like!
Jacob Hoover: This is an absolute gem of a problem.
Ian Shanahan: Classic half-battery involving numerous tries, with the black queen losing focal control of each of the two battery lines in turn thence being shut off by one of the batteries’ firing pieces for her troubles (the Mackenzie theme). A ‘trying’ problem, difficult to solve, but flaunting rich play and numerous changes between try and key.

If it’s Black’s turn in the diagram, every legal move gives mate immediately. White cannot avoid disturbing this block position, however; the white queen in particular has numerous tries that are subtly defeated. 1.Qa5/Qb4/Qc5/Qc4? (controls e1 or c1) Sa3+! (not 1…Sc3, a shut-off mate), 1.Qa4? (controls d1) Sc3+! 2.Qd1+ Sxd1 (not 1…Sa3+? 2.Qd1+ Rxd1), and 1.Qd7? (controls h3) Qxh3+! 2.Qxh3 Bg2+ (and ironically White is forced to mate with 3.Qxg2). Most rook moves along the h-file are refuted by the piece’s capture by the black queen, but 1.Rh7! (with a technical threat of 2.Rh3) works by drawing the queen to the right square for 1…Qxh7 2.Qd3+ Qxd3. Other vertical defences by the queen are answered by a grab – 1…Qh6 2.Rxh6, 1…Qh5 2.Rxh5, 1…Qh4 2.Rxh4 (and 1…Qh3 2.Rxh3), all leaving Black in zugzwang and compelled to mate with 2…S~/Bg2/g2. Since the set play totally consists of immediate mates by Black and the key has to extend the play to two moves, this selfmate is a pseudo one-mover (not possible in a directmate).

Andy Sag: All black moves are mates in the set position. The tricky key forces the black queen to move but stay on the h-file. Many tries, notably 1.Qd7? Qxh3+! and 1.Rxh2? gxh2! and the black king is free to move to g3.
Jacob Hoover: The key must give Black at least one non-mating move and 1.Rh7! does this quite nicely.
George Meldrum: A nice hair-pulling problem with lots of ‘almost works’ solutions. The line that ends with the black queen moving to d3 to mate the white king is absolutely brilliant.
Andrew Buchanan: Very clever. I like the geometry of this very much, and the solution is well hidden.
Ian Shanahan: A cleverly constructed problem! Any move by the white queen fails.
Bob Meadley: That's a lovely selfmate by Mangalis.

The key 1.Qe6! threatens a pin-mate, 2.Qc8, against which Black can defend by unpinning the g4-queen. The two black pieces capable of doing this are themselves half-pinned by the white bishop, so when either moves, the other is immobilised and this leads to another pin-mate: 1…Be4 2.Rd4 and 1…Sf4 2.Qe4. Remarkably, both variations involve another strategic effect: the defender interferes with the black queen’s control of the mating square on the fourth rank. Two other defences unpin the d5-knight, which gives a pair of battery mates, nicely differentiated: 1…Rg6 2.Sf6 and 1…Bf5 2.Se3. The by-play makes additional use of the thematic pieces: 1…Sd4 2.Bxd3 and 1…Rxd5+ 2.Qxd5.

Andy Sag: Pin-mate theme with many tries: 1.Qg8? Sf4!, 1.Qe8? Rxd5+!, 1.Qf3? Sc1!
Ian Shanahan: Classic Good-Companions-style problem from the Italian expert. In the thematic variations, each half-pinned piece unpins by interference the black queen; pin-mates result. Rich strategy indeed!

The waiting key 1.Kf4! completes the block. A random move of the d8-rook unguards d7: 1…R~ 2.Qd7, while its correction move blocks a flight: 1…Rd6 2.Qe4. Similarly, the g8-bishop unguards f7 with a random move: 1…B~ 2.Bf7, and self-blocks with a correction: 1…Be6 2.Qc5. The a5-knight provides a third pair of thematic defences: 1…Sa~ 2.Bc6 and 1…Sc4 2.Rd3, involving the same sorts of errors as the other pairs. Lastly, 1…Sg~ enables 2.Qe5. A textbook example of correction play, attractively set.

Andy Sag: An easy to solve pawnless, Meredith, dual-free waiter with three self-block and four unguard variations.
George Meldrum: Three of Black’s pieces have moves for which White has no reply. The key move covers e4 and e5 enabling the zugzwang setup.
Andrew Buchanan: Mirror mates abound in this aristocratic position. For three of the black pieces, there is a generic move, plus a special one which blocks a square adjacent to the black king and allows a different mate. The key is a bit aggressive, but I can't see any interesting alternative diagram.
Jacob Hoover: A nice black-correction problem, and in Meredith and aristocrat to boot.
Ian Shanahan: The waiting key leads to secondary corrections by three of the black pieces in an economical setting. A beautiful, subtle problem – as one would expect of its author.
Michael McDowell indicates that this problem shows correction play only once, not three times! He writes: “Only the knight shows correction play. 1…Sc4 commits the random error of unguarding c6, corrects by closing the line c3-c6 and commits the second error of blocking c4. Neither 1…Rd6 or 1…Be6 commit their random error of unguarding d7 or f7. They simply avoid making it, so they don't correct anything.”

The key 1.Qb4! pins the f4-pawn and threatens a short mate, 2.g3. In the thematic variations, Black employs the c8-rook and a6-bishop to interpose on the fourth rank and unpin the pawn, but these defences result in a mutual interference between the two pieces. 1…Rc4 2.0-0 – vacates e1 to threaten 3.Be1 – 2…Rxc3/f3 3.g3 or 2…Rg6 3.Sxg6. 1…Bc4 2.0-0-0 – again threatens 3.Be1 – 2…B~/f3 3.g3 or 2…Rg6 3.Sxg6. Thus a black Grimshaw on c4 sets off white castling on both sides – very harmonious. There’s good by-play in which Black also defends by unpinning the f4-pawn: 1…Sd4 2.Qe7+ Rf6 3.Sg6 or 2…g5 3.Qxg5; 1…e4 2.Qxe4 (threat: 3.g3) Rg6 3.Sxg6.

Andy Sag: King and rooks in original positions make second-move castling an obvious suspect. Denis was well known for using castling in his problems. The threat is a short pin-mate with four pin breaker defences. The pin-mate after 1…Sd4 2.Qe7+ Rf6 is a nice touch.
Dennis Hale: The point of the problem is to activate a Grimshaw based on c4 with the two unpinning defences, 1…Bc4 and 1…Rc4. It is pleasing how each of the two self-interferences is met by 0-0-0 or 0-0, but not both.
Jacob Hoover: One try 1.Qa4? has the same idea as the key but dies to 1…Sd4! since from a4 the queen has no access to the e7-h4 line.
George Meldrum: Even though it was obvious that castling was to be a feature the need was obscure. Lovely spread of pieces and ingenious execution of interference play.

Two important set variations are 1…Qxd4 2.Sxf4 and 1…Be6 2.Qc6. The first of two thematic tries, 1.Bc5?, threatens one of these set mates, 2.Sxf4, and the other set mate recurs but against a new defence: 1…dxc5 2.Qc6 (1…R~ 2.Bxe4, 1…Bh6 2.Qxg8). This try is refuted by 1…Qxd4! (a paradox as this defence allows 2.Sxf4 in the set play but defeats it here). The second thematic try 1.Rxf6? threatens the other set mate, 2.Qc6, and now another new defence induces the first set mate: 1…exd4 2.Sxf4 (1…Rxf6 2.Bxe4). But there’s no answer to 1…Sb4! Besides rendering a double mate transference relative to the set play, the two tries show the pseudo le Grand pattern: a reciprocal change of threat and variation mate (the latter in response to different defences). The key 1.Sxc4! grants a flight on c4 and threatens 2.Sb6. Now the two thematic mates appear again but they work against a third pair of defences: 1…bxc4 2.Sxf4 and 1…Kxc4 2.Qc6. Also, 1…Be7 2.Qxg8. An amazing 3x2 mate transference scheme, blended with the pseudo le Grand.

Andy Sag: The key allows a flight-capture. Curiously, the threat becomes a pin-mate after 1…Qxd4 and duals with 2.Rxd4, another pin-mate, and one of the variations is also a pin-mate.
Jacob Hoover: In this problem we have mate transference across four phases of play. Each try transfers one of the thematic mates to the threat while transferring the other to another defense. Also, the key transfers both mates to completely different defenses altogether (that both of these defenses are captures at c4 was a very nice touch!).
Ian Shanahan: It was difficult to distinguish between the symmetrical try and the key. A heavy but satisfying problem.

In the set play, the white queen gives two different mates when unpinned by the black knights: 1…Sd3 2.Qd4 (not 2.Qxb4?) and 1…Sf5 2.Qxb4 (not 2.Qd4?). Another set variation is 1…e5 2.Bd6. The key 1.Sa5! places an extra guard on c4 and threatens 2.d4. The knight defences of the set play are effective against the threat, but White must exploit the unpins in new ways since the key-piece has also unguarded d4 and b4. 1…Sd3 2.Qe3 (not 2.Qe5?) and 1…Sf5 2.Qe5 (not 2.Qe3?). Another changed mate occurs with 1…e5 2.Bxb6, utilising the opening of the sixth rank in both directions. Lastly, 1…Rd6 permits 2.Sb7. Three good changed mates, two with dual avoidance effects.

Andy Sag: The key abandons set mates after two queen unpins and sixth rank line clearance. The fourth defence 1…Rd6 unguards b7 and also refutes the try 1.Se5?
George Meldrum: The line 1…Sd3 2.Qe3 is particularly nice.

Two potential mating squares for the black king are a6 (for Ra3) and c8 (for Rh8), neither of which is accessible unless the white knight is captured. But if the black knight makes such a capture on b7, the piece finds itself preventing either rook mate and cannot move again without discovering check. The actual plan is quite different: the king is to be mated on its original square by the h5-rook along the fourth rank. This requires Black to remove both white pawns, and the one on d4 cannot be captured by the knight because of check. Therefore the black king has to make that capture before returning to a4. Such a trek by the king involves its multiple shielding by the knight against the c3-rook.  1.Sc1 2.Sb3 3.Ka3 4.Kb2 5.Sa1 6.Sc2 7.Kc1 8.Kd2 9.Se1 10.Sd3 11.Ke3 12.Kxd4 13.Ke3 14.Kd2 15.Se1 16.Sc2 17.Kc1 18.Kb2 19.Sa1 20.Sb3 21.Ka3 22.Ka4 23.Sd2 24.Sf3 25.Sxh4 Rxh4.

Jacob Hoover: If Black were to remove both white pawns, White could mate with Rh4.
Andy Sag: The king is confined to ranks 1 to 4, so the knight must do a merry dance avoiding checking to allow the king trip to d4 and back.
George Meldrum: Nice!
Andrew Buchanan: Quite clean and enjoyable.

GM Miodrag Mladenović proposes an interesting extension of the problem, shown below.

Now there are four thematic captures to clear the fourth rank. 1.Sf2 2.Sxg4 2.Se3 4.Sxc4 5.Sd2 6.Sb3 – transposing to the original solution – 7.Ka3 8.Kb2 9.Sa1 10.Sc2 11.Kc1 12.Kd2 13.Se1 14.Sd3 15.Ke3 16.Kxd4 17.Ke3 18.Kd2 19.Se1 20.Sc2 21.Kc1 22.Kb2 23.Sa1 24.Sb3 25.Ka3 26.Ka4 27.Sd2 28.Sf3 29.Sxh4 Rxh4. Miodrag likes how the knight starts from the h1-corner and during the solution visits the a1-corner twice.

Set mates are prepared for most of Black’s moves in the diagram, except for those by the d4-bishop that retain control of e3, stopping a queen mate there. The key 1.Kh3! (waiting) completes the block by giving the queen access to g2. 1…Bf2/Bg1/Bc5 2.Qg2 since the b2-bishop is allowed to guard the e5-flight, and 1…Bc3/Bxb2 2.Qe3 (with duals following other black bishop moves). 1…f4 opens a queen-line to e5 and enables 2.Bh7. Taking the flight produces 1…Kxe5 2.Re6, and lastly 1…S~ 2.Qf4. Not 1.Kf1/Kh1? Sg3+! For part (b), 1.Kh3? fails to 1…f4! The new key is 1.Sd7! (waiting), which removes the e5-flight and opens the e-file. Now the black bishop shows focal play: 1…Bf2/Bg1/Bb6/Ba7 2.Re6, 1…Bc5 cuts off the b5-rook and prevents 2.Re6, but permits 2.Sxc5, and 1…Bc3/Bxb2 2.Qe3 (again duals follow other bishop moves). Therefore two changed mates occur between the twins for 1…Bf2/Bg1 and 1…Bc5, and a third appears with 1…f4 2.Qxg6. The by-play 1…S~ 2.Qf4 is the same. In part (a), 1.Sd7? is refuted by 1…Bc5! An intricate waiter with three clever changes.

Andy Sag: Interesting twin but both parts have three duals and the key in part (b) takes a flight, all of which spoils the problem somewhat.

The solution of part (a) runs 1.Rxd3+ Re5+ 2.Kd4 Re4. Both players have a diagonal battery aimed at the opposing king, and the black rook fires one immediately while capturing a knight. The rook, in doing so, blocks a flight on d3 and vacates d4 for the king. White in reply opens the battery too with the rook, producing a cross-check. This move self-pins the rook on e5, but the piece duly mates after it gets unpinned by the black king on d4. Part (b), solved by 1.Rxd5+ Se5+ 2.Kd4 Sc6, shows many corresponding effects. The black rook again commences play by firing the battery but captures the rook instead, to self-block on d5 and vacate d4. Now the white knight interposes on e5 and opens a new diagonal battery (formed by the twinning), for another cross-check. Finally, the black king moves to d4 and unpins the knight, which then mates. The Zilahi theme is featured since White’s rook and knight swap their functions of getting sacrificed and delivering mate.

Jacob Hoover: In each solution we have, in order, a discovered check, a cross-check, an unpin, and a mate by the piece that was just unpinned.
Andy Sag: The h8-bishop is necessary in part (b) to make the knight go via e5. The f5-pawn stops a cook in (b) ending with the white rook on f5 and knight on f2.
Thomas Thannheiser: Very nice center dance of the black and white pieces!
George Meldrum: Magnificent, unexpected and pleasing solution.
Michael McDowell: Michal Dragoun’s helpmate features a good illustration of Chris Feather’s “weasel”, a piece which adds strategy but is not strictly needed to make the solution work. Replace the pawn on c5 with a white one on b4 and the black bishop can be removed.

The black king has an unprovided flight on b6 and the only way to deal with it is 1.Qe1!, after which 1…Kb6 does not stop the threat of 2.Qb4. But the main point of the key is that it grants two flight-captures while sacrificing the key-piece. 1…Kxd6 admits a pin-mate, 2.Qe7, and 1…Kxd4 enables 2.Qc3. Taking the offered queen allows White to activate the B + R battery: 1…fxe1=Q 2.Rxd5. Lastly, 1…Qxd4 blocks one of the conceded flights and opens a white rook line to the other, permitting 2.Qa5. This 19th century problem incorporates a wonderful key and one move it provokes, 1…fxe1=Q, produces the maximum possible change in terms of material balance!

Jacob Hoover: The key sacrifices the queen, and passively sacrifices both the d4-rook and d6-knight. The threat is effective after 1…Kb6, but since the king is mated on a different square one could argue that that mate is distinct.
Andy Sag: I knew the queen had to move to allow 1…Kxd6 and the g1-bishop made the key fairly obvious.
Atagün Karayel: After looking for a purpose of f2-pawn, it appeared right before my eyes. Many cute mates follow.
George Meldrum: Somewhat strange problem but not complaining when a key adds another two king flights!

Only two black moves in the diagram, 1…Se7 and 1…Sf6, are not provided with a set continuation. After the subtle key 1.Kf7! (waiting), these knight moves are exploited as shields for the white king against prospective checks. 1…Se7 2.Sb4 – White unpins the d2-rook but since 2…Rd7 no longer checks, Black cannot stop the threat of 3.Rxa6. Similarly, 1…Sf6 2.Rb4 sees the black rook unpinned but by a different white piece, and with 2…Rxf2 neutralised, there’s no defence against 3.Rb5. The remaining play proceeds as set. 1…a2 2.Sb2 and 3.b4, 1…b5 2.b4+ Kxa4 3.Sc5, 1…g5/gxh5 2.Rf5+ b5 3.Rxb5, and 1…Sf~ 2.Bxd2.

Andy Sag: The g8-knight is the only mobile black piece not tied to a critical guarding role. The key ensures that knight can only move to a position enabling White to unpin the black rook but leave it unable to check.
Jacob Hoover: Two thematic tries threaten immediate mate but unpin the black rook too soon: 1.Rb4? (threat: 2.Rb5) Re2+!; 1 Sb4? (threat: 2.Rxa6) Re2+!
George Meldrum: The black knight on g8 prevents tries like 1.Rf7? and 1.hxg6? The brilliant key move uses the black knight moves to defend from checks giving the knight kudos which ironically ends in its self-defeat.
Michael McDowell: A very clearly constructed incomplete block with two beautifully matched variations added where knight shields allow unpins of the rook. The clue for the solver lies in determining the purpose of the seemingly superfluous pawns on the h-file.

Black has a strong defence without a set reply in the diagram position, 1…Qxf6, which provides a hint to the solution. The key 1.Qd6! (double-threat: 2.Rg6/Rxf7) allows the black pawn to capture the queen with check. This e7-pawn actually has four defences that yield different mates, producing the BP4 theme: 1…exd6+ 2.Rxd6, 1…exf6 2.Qxf6, 1…e5 2.Rg6, and 1…e6 2.Rxe6. All four mates necessitate the pin of the black queen, which swaps roles with the pawn when it plays the strong defence mentioned – 1…Qxf6 2.Qxf6 – the move self-pins the pawn and hence completes the half-pin on the 7th rank. The by-play largely repeats white mates already seen: 1…Qe6 2.Rxe6, 1…Qg6/Rf8 2.Rxg6, and 1…Rxh7 2.Rxf7, where the threats are separated in the last two variations. Two traditional themes are attractively integrated in a setting with no white pawns plus a good key.

Andy Sag: Tries 1.Qb6? Bc6! and 1.Qxe7? Rf8! The key sacrifices the queen with check. Good demonstration of pin-mates, battery mates and double-checks. The 7th rank half-pin and diagonal battery are strong clues.

Black’s potential mating move …Kb7 (covering the a6-flight) is stopped by both the white queen and the white knight. If the latter unguards the diagonal battery, White will threaten 2.Qb7+ Kxb7. The key is 1.Sd2!, after which the c5-pawn has two capturing defences. 1…cxd4 grants the black king a flight on d6, but because the black pawn is now immobilised, White can play 2.Se4, which controls d6 and also interferes with the queen on the long diagonal, and forces 2…Kb7 through zugzwang. 1…cxb4 defends by granting the white king a flight on b4, and again the black pawn gets blocked; now 2.Rd5 opens the black rook’s line to b4 and closes the queen’s diagonal, forcing 2…Kb7 through zugzwang once more. Bob Meadley wrote in W. A. Whyatt’s Chess Problems, “Bill’s first foray into the self-mate world results in a 4th prize.”

Andy Sag: It is clear that the knight must move to stop it from going to b6. To force 2…Kb7, White must unguard b7 and keep a guard on d6 and in the case of 1…cxb4, use the pinned rook to stop 3.Kxb4.
Nigel Nettheim: In retrospect, the b3 and d3 pawns are lying in wait for either capture from c5, but one might not have noticed that at first. Relatively easy.

The black king has three flights, of which one is provided: 1…Kxe4 2.Qh1. After the key 1.Qxb2!, which threatens 2.Qxd4, the flight defences lead to a variety of mates: 1…Kxe4 2.Qg2 (a concurrent change from the set), 1…Kxc4 2.Rxc7, and 1…Kc5 2.Qb5. Black also self-blocks on each of the flight-squares at least once: 1…Rxe4 2.Rf5, 1…Rdxc4 2.Qe5, 1…Rcxc4 2.Re5, and 1…Bc5 2.Rfe7. Harmonious defence motifs, and in response the white queen delivers three mates while the B + R battery opens three times.

Andy Sag: The key must provide for two unprovided flights to the c-file.
Michael McDowell: I draw your attention to the improved version of the Sun-Herald two-mover which Touw included in his collection [shown below]. Not a bad problem for a teenager!

In the solution where White mates, 1.Ka6 a4 2.Sb7 Ra8, the black king goes to an edge square and a black piece follows to self-block, leading to a sideboard model mate. When the roles are reversed and Black mates, the solution 1.Kxh7 Sf5 2.Rg8 Rh6 shows similar effects: the white king’s move is followed by a self-block on the piece’s initial square, to set up another sideboard model. An economical duplex problem that isn’t too easy to solve.

Andy Sag: Clever miniature with two problems in one setting. The h7-pawn is there to stop a similar mate with the king on h8 instead of h7.
Andrew Buchanan: If all the pieces are shifted down one row, the black pawn can be removed.

Set defences by the black queen, including some prominent checks, mostly induce the R + S battery to open: 1…Qa5+ 2.Sxa5, 1…Qxb8 2.Sxb8, 1…Qxc6+ 2.Rxc6, and 1…Qxe5+ 2.Sxe5/Sd4. The key 1.Qe8! (threat: 2.Qg8) leaves the c6-knight guarding e5 on its own and disables three of the set mates. The newly formed Q + B battery is used to handle the queen defences instead: 1…Qa5+ 2.Bb4, 1…Qxb8 2.Bd8, and 1…Qxc6+ 2.Bc5. The set dual is removed in 1…Qxe5+ 2.Sd4. 1…Qc8/Qd8 also permits 2.Bd8. Lastly, similar to the set 1…Qxc6+ 2.Rxc6, 1…Qxe7 2.Qxe7 utilises the thematic pieces in a simple recapture. Good cross-checks in both set and actual play.

Andy Sag: The key sets up an additional battery and five changed mates [including changes from the set duals, 1…Qd8 2.Sxd8/Sd4 and 1…Qxe7 2.Sxe7/Sd4/Sd8] in a Meredith setting. The unpinning try 1.Kd2? Bf5! has plenty of play including cross-checks but fails to a lowly bishop move.
Jacob Hoover: 1…Qxc6+ is also a self-pin, and 1…Qxe5+ unpins the knight while self-blocking. The black bishop seems superfluous, but its presence defeats attempts to unpin the knight directly: 1.Bc5/Kd2? (threats: 2.Sd4/Sd8) Bf5!
Ian Shanahan: Lovely changes of mate (including the responses to all three checks) between set and actual play. A worthy prize-winner!

The key 1.Rg2! contains two technical threats, 2.Sbc3 and 2.Sdc3, that are prevented by virtually all black moves. The black pawn has two captures, each of which separates into a queen and a knight promotion, to generate four distinct variations. 1…cxb1=Q 2.Qc4+ Qc2 3.Qxc2 or 2…Kxd1 3.Qf1. 1…cxb1=S 2.Qa1 Kxd1 3.Qxb1. 1…cxd1=Q 2.Qa3+ Kxb1 3.Qb2. 1…cxd1=S 2.Qf1 Kxb1 3.Qxd1. After 1…Kxb1, the far from obvious 2.Qa4 puts Black in zugzwang, though some duals follow: 2…c1=Q 3.Qa2, 2…c1=S 3.Rb2/Sc3, 2…cxd1=Q 3.Qxd1, and 2…Kc1 3.Qa1/Qxc2. 1…Kxd1 results in a short mate, 2.Qf1/Rg1. An active white queen delivers a pleasing assortment of mates. The two knight promotion defences lead to symmetrical play but it’s not a big flaw to repeat a good variation!

Andy Sag: Neat miniature with different mates for promotions to queen and knight on three different squares. One short mate but plenty of variety in play.

In this complete block position, the black king has a provided flight: 1…Kd5 2.Bb7. Random moves by the black knight enable the same bishop mate, 1…S~ 2.Bb7, while the self-blocking correction 1…Sd5 admits 2.Bd3. White has no way of preserving all of the set play, e.g. 1.Bb5? Kd5!, 1.Bc8? Sd5! Two other good tries generate a threat: 1.Rb5? (2.Re5) Sd5!, and 1.Bf1? (2.Bg2) Kd5! The key 1.Sa4! surprisingly also involves a threat, 2.Sc3. The resulting variations show two changes from the set: 1…Kd5 2.Sc3 and 1…Sd5 2.Sc5.

Andy Sag: Position is a complete block but no white move can maintain the set mates. The key introduces a threat and the set bishop mates are replaced by knight mates.
Atagün Karayel: The key 1.Sa4! gives up support for the set-play mate, Bd3.
Brian Stephenson: A classic block-threat.
Jacob Hoover: The threat still is effective after 1…Kd5, although this is arguably a distinct mate due to the king being mated on a different square. A threat-mutate, with bishop mates in the set play replaced by knight mates in the actual play.
Andrew Buchanan: A delightful and harmonious block-threat.
Nigel Nettheim: A complete block, and all set variations are changed. But Black’s king is exposed to White’s overwhelming force so a mate is not surprising, and I find it neat rather than sparkling.
Ian Shanahan: A sweet and simple block-threat, with two mates changed. It's a pity that the mating move after the king-flight is the same as the threat. Constructionally, the e-file pawns stop 1…Sd5 2.Re6. Getting rid of them and replacing them with a black f7-pawn is no good, because then 1.Rxf6 cooks the problem.

An en passant capture of the c5-pawn is justified by the following retro-analysis. White’s pawn structure requires five captures and Black’s requires two, accounting for all missing units. Black’s last move couldn’t have been:

• …c6-c5/Kb7-b6/Rh5-h4 due to impossible checks.

• …Kc7-b6 since the uncheck e5xd6+ adds two more white pawn captures.

• …g7xh6 because the original bishop from f8 must have come out earlier to be captured by a white pawn.

• …f3xe2 because such an immediate uncapture would leave this black f-pawn behind White’s original f-pawn now on f4 or f7, and with no spare units to be captured by either side, the two pawns couldn’t have gotten around each other. That is, a white pawn must retract to f2 before Black could make the uncapture, …f3xe2.

The only possible last move by Black is therefore …c7-c5, which legalises the key, 1.bxc6 e.p.! White has four threats utilising the B + S battery, 2.Sb5/Sf5/Sf3/Sxe2, and each is forced individually with 1…exd6 2.Sb5, 1…Rh5+ 2.Sf5, 1…Rxh3 2.Sf3, and 1…e1=Q 2.Se2 – the Fleck theme. Black can parry all of the threats but these total defences bring about new mates: 1…dxc6+ 2.Sxc6, 1…dxe6+ 2.Sxe6, 1…Sxb3 2.Sxb3, and 1…Sxc2 2.Sxc2. Since the four additional mates match the number of separated threats, a Fleck-Karlstrom is demonstrated. Furthermore, the eight mates executed by the d4-knight produce a knight tour. The thematic knight is captured in the set 1…cxd4 2.Bxd4, but after the key there are no extraneous variations.

Andy Sag: A retro with a complete knight wheel (or tour?).
Jacob Hoover: We have the Fleck theme; the other four defenses are square-clearances. Since all eight possible moves of the d4-knight are seen and each one is the response to a different defense, we have a white knight wheel.
Nigel Nettheim: This problem shows a knight wheel and a retro. Either of them separately would be excellent, but the combination of the two is nothing short of amazing!
Dennis Hale: A first-class problem and a worthy 500th.
Andrew Buchanan: What a fabulous idea, superbly executed, for the 500th problem! It's amazing how much design space exists just by harmoniously combining two different themes, hitherto not associated.
Ian Shanahan: It is very rare to have such rich post-key play (i.e. the Partial Fleck theme and a white knight wheel) in a two-mover with such rich retro-analytic content. An amazing problem, and I really do hope it wins a prize.

Other successful solvers were Karel Hursky, Bob Meadley, and George Meldrum. Well done!

White has three potential mating moves on e5, c5, and b4, each of which is guarded by a black pawn. Since these pawns are blocked, White aims to divert them with a sacrifice. Meanwhile, Black must commence play and the only free unit available is the knight, which serves no positive function. In each solution, the piece makes a waiting move and finds itself controlling two of the three mating squares. Thus White needs to proceed carefully with the appropriate sacrifice, one that leads to mate on the sole unguarded square. 1.Sa6 g5 2.fxg5 Se5, 1.Sc6 a5 2.bxa5 Rxc5, and 1.Sd7 exd4 2.cxd4 Rb4. Black tempo play is pleasantly combined with dual avoidance.

Andy Sag: The black knight has three possible tempo moves, each guarding two of the three possible mating squares followed by pawn moves to complete the unguard.
Jacob Hoover: Each solution has the black knight moving to put an additional guard on two of the three potential mating squares while not adding a guard to the third.
Nigel Nettheim: The black knight must keep out of the action, and the ways in which it does so form a nice trio. The e4-pawn prevents some mates with cxd3.