Weekly Problems 2024-B

After the key 1.Sh5!, the two white knights can carry out four similar threats that are sacrificial deflections: 2.Sf4+ [A] Sxf4 3.Se3 [B], 2.Se3+ [B] Sxe3 3.Sf4 [A], 2.Sxf6+ [C] Sxf6 3.Se7 [D], and 2.Se7+ [D] Sxe7 3.Sxf6 [C]. The AB-BA and CD-DC labels show an incidental reversal pattern of White’s second and third moves. The main theme, however, consists of how Black compels White to play the four second-move threats – A, B, C, and D – one by one. 1…c3 creates a flight on c4: 2.Sf4+ [A] Kc4 3.Be6, 2…Sxf4 3.Se3 [B]. 1…Rxc8 creates a flight on e6: 2.Se3+ [B] Ke6 3.Sg7, 2…Sxe3 3.Sf4 [A]. 1…Rxb5 creates a flight on c6: 2.Sxf6+ [C] Kxc6 3.Bd7, 2…Sxf6 3.Se7 [D]. 1…e3 creates a flight on e4: 2.Se7+ [D] Ke4 3.Bf5, 2…Sxe7 3.Sxf6 [C]. The Fleck theme, or separation of the four white threats, is wonderfully combined with star-flights. Skilful construction is illustrated by the white bishop on c8, which apparently serves to guard e6 but ends up also delivering three different mates.

Andy Sag: Post key, all four knight checks are effective threats exploiting the overloaded black knights. Each of the four defensive lines creates a different diagonal flight but in each case one of the knight checks provides for the flight as well as activating one of the original threats.
George Meldrum: The multiple threats are separated by Black’s play which includes giving four king flights.
Jacob Hoover: We have the three-mover analogue of the Fleck theme (which is primarily seen in two-move directmates). In addition – and I found this to be quite the nice geometric touch – after the key the four knights make a rhombus!
Rauf Aliovsadzade: A masterpiece!

Initially, Black has a number of checks that are provided with set mates: 1…Qf8+ 2.gxf8=S and 1…S8~+ 2.g8=Q. In both cases (except for 1…Sf6+ allowing a dual, 2.g8=Q/S), White must choose the right promoted piece to give a double-check(mate), because the Q + P battery is controlled by the black bishop. The key 1.Sd4! opens a line for the white bishop and threatens 2.Qxg6. Since the key-piece has granted a flight on h6 to the king, most of the set replies to the checking defences no longer work, but given that the knight has also cut off the black bishop, new battery mates – aimed at recovering h6 – are possible. 1…Qf8+ 2.gxf8=Q and 1…S8~+ 2.g8=S. If 1…Sh6+ attacking the queen, then the self-block enables 2.g8=Q as set. The flight-move 1…Kh6 does not counter the threat, 2.Qxg6. Against the two thematic defences, White promotes on the same squares as in the set play, but the choice of promoted pieces is reciprocally changed.

Andy Sag: All black defences involve checks answered by various pawn (sub) promotions. Pity the setting could not be reduced to 12 pieces.
George Meldrum: No way the white knight can make the first move… Wait, what?!

A rook mate on the g-file seems likely in view of how the king is restricted. Any black unit blocking f5 would also disturb the mate, so the white king has to be utilised to guard that flight. The king needs four moves to reach e4, meaning the rook has no spare moves and it will deliver mate on g2 in one step. Therefore Black must clear the second rank by promoting all four pawns, without the new pieces attacking the g-file or hindering the white king’s trek. Black’s remaining move is allocated to disabling the c8-bishop’s control of g4. 1.f1=R Kb4 2.b1=R+ Kxc3 3.c1=R+ Kd3 4.d1=R+ Ke4 5.Rd7 Rg2. Four black rook promotions are precisely forced and the task is achieved with good economy of length and a model mate.

Andy Sag: Black pawns promote to rooks in the correct sequence to allow the white king to escape a series of checks and navigate through a minefield to e4.
Jacob Hoover: The order of the promotions is carefully chosen so as not to ruin the plan, as are the promotions themselves – all of them to rooks!
Andrew Buchanan: With so many moves, there are potentially many squares where the black king can be mated on, even though White only has one unit. It surprises me that a sound setting can be found.
George Meldrum: Brilliant and funny too.

If it were Black’s turn in the diagram, only the g1-knight is free to move and this would result in an immediate mate. White cannot maintain the block position, however, and the solution will extend the play to two moves. In two thematic tries and the key, White releases each of the three black pieces pinned on the b-file, only to capture it after its various moves, in order to force the set mate 2…Sg~ by zugzwang. 1.Qb6? unpins the knight: 1…Sa3 2.bxa3, 1…Sxc3+ 2.bxc3, 1…Sxd4 2.cxd4, 1…Sa7 2.Qxa7, but 1…Sc7! refutes since 2.Qxc7 is mate. (1.Qb7? Sc7 2.Qc8, but now 1…Sxd4!) 1.Ra3? unpins the bishop: 1…Bxa3 2.bxa3, 1…Bxc3 2.bxc3, 1…Ba5 2.Rxa5, but 1…Bc5! refutes. The key 1.Bb1! (waiting) unpins the rook: 1…Ra3 2.bxa3 S~, 1…Rxc3 2.bxc3 S~, and 1…Rxb2 2.Bxb2 S~. The Grab theme, in which White captures the same black piece on multiple squares, is presented thrice, and in all three phases the white play is completely dual-free.

Andy Sag: As set, Black can only move the g1-knight giving a battery mate but any white move will either mate or allow one or more additional black moves. The key must provide for the newly mobilised black unit to be captured, pinned or otherwise immobilised (without checking), again leaving only the g1-knight mobile.
Jacob Hoover: A pseudo-one-mover. Moving either knight or the c3-pawn gives Black too much freedom; therefore White must unpin one of the pinned black units. This one was moderately difficult.

After the try 1.Sf3? (waiting), the four black pieces on the top rank are each defending against a mate, and they cut off one another when attempting to preserve their guards. 1…Be7/Be6 2.Re5, 1…Re6/Rf7 2.Rxc4, 1…Bf7/Bf6 2.Bxf5, and 1…Rf6/Re7 2.Sfg5. These variations show four pairs of Grimshaw interferences. Additional play is 1…c3 2.d3 and 1…f4 2.Sf2, but Black thwarts the try with 1…Bd5! The key 1.Rxf5! (which self-pins the rook on the f-file) is again a waiting move; now four new mates arise from the same self-weakening defences. 1…Be7/Be6 (1…Re5) 2.Qe5, 1…Re6/Rf7 (1…Bd5/Bh7) 2.Qd5, 1…Bf7/Bf6 2.Rf4, and 1…Rf6/Re7 (1…Bg5/Bc7, etc.) 2.Sg5. Also, 1…c3 2.d3 and 1…Rxf5+ 2.Qxf5/Bxf5. The arrangement of the two pairs of black rooks and bishops – leading to mutual interferences between them – is called organ pipes. The idea originated with Sam Loyd in the 19th century, whose best-known presentation is P1019649. The current problem gives the scheme a modern touch by incorporating a thematic try, which effectively doubles the content.

Andy Sag: The Bristol-style key allows a check and poses no threat. Unfortunately there is a dual after the check. The problem appears to be based on one composed by Sam Loyd in 1859, which was republished in The Weekend Australia last Saturday [the same day as this Weekly Problem, by a remarkable coincidence].
Bob Meadley: Looks like 1.Rxf5 waiting and the organ pipes get blocked in trying to find a relief move.

The black king has flights to g5 and h5, while the initial pin of the white queen by the black one is another impediment (which disables cooks like 1.Ba2 hxg6 2.hxg6 Qxg6). If White tries to guard h5 with 1…Bxg6, a queen mate on d2 cannot be arranged given the four black pieces converging on the mating line. Instead, Black aims to move the g6-knight to activate the white bishop’s guard, and also to cut off the d8-bishop. Black thus begins by unpinning the knight, using the queen to do so as that would simultaneously release the white queen. 1.Qb6+ Qc5 2.Se7 Qe3. To answer the black check, the white queen immediately re-pins itself to its counterpart, only to give mate by travelling along the diagonal pin line – a mate that exploits the black queen’s own pin by the rook. In part (b) with the e5-pawn removed, the first solution fails as the d5-rook controls g5. The twinning, however, has also given the black queen access to f6, where it could unpin the g6-knight and interfere with the d8-bishop, such that the knight is free to shut off the d5-rook instead. 1.Qf6+ Qf5 2.Se5 Qf4. A different black check compels another self-pin by the white queen, which then moves along the pin line – now orthogonal – to deliver mate, one that also relies on the pin of the black queen.

Andy Sag: Well matched twin featuring a feast of pins and unpins and ending with pin-mates from pinned positions. How cute is that!
George Meldrum: I love it!

The composer writes, “Several years ago Mike Prcic published an article in StrateGems that dealt with the Lincoln theme being used in three-movers. Lincoln theme: at least three variations have the same White’s 2nd move but end with a different checkmate.”

With 1.Rd8! the key-piece vacates d4 (and keeps an eye on f8) to threaten 2.Rf4+ [A] Kxe5 3.d4 [B]. In the two main variations, White still plays 2.Rf4+ but the follow-up mates are different, so if we include the threat in the scheme, the Lincoln theme is brought about. 1…Bxb2 2.Rf4+ Kxe5 3.Bxb2 and 1…Bxd3 2.Rf4+ Kxe5 3.Sxd3. Good supplementary ideas include an AB-BA reversal of White’s moves compared with the threat sequence: 1…Kxe5 2.d4+ [B] Kf5 3.Rf4 [A]. The black knight illustrates correction play with 1…S~ 2.g4+ Kxe5 3.Sf3 and 1…Se4+ 2.dxe4+ Kxe5 3.Sf3. Lastly, 1…fxe5 2.Rf8+ Sf7 3.Rxf7 and 1…Bd7 2.Sxd7 (threats: 3.Rf4/g4) Se4+ 3.dxe4. The three thematic lines – and the black knight variations as well –  involve a curious form of changed mates against the same defence, 2…Kxe5.

Andy Sag: The key must provide for 1…fxe5 which limits the number of moves to analyse.
George Meldrum: The key opens new variations, including play for the unfriendly 1…fxe5.

Each side is missing a bishop and a pawn, one of which was sacrificed on the b-file to account for the doubled pawns. It’s more efficient to sacrifice a bishop on b6/b3, after the piece has captured the neighbouring a- or c-pawn on its original square. The first solution starts symmetrically with 1.d4 d5 2.Bf4 Bf5 3.Bxc7 Bxc2 (thus not 1.d4 d6? when the white bishop is blocked from both c7 and a7) 4.Bb6 Bb3. Black must quickly release the white rook, which only has the 6th rank available to reach the king-side. Not 4…axb6? 5.Qd3 Bb3 6.axb3 b5 7.Ra6 Qa5+ when the check prevents 8.Rg6 Ra6. 5.axb3 axb6 6.Ra6 b5 7.Rg6 Ra6 (7…Qa5+? 8.Sd2 Ra6 9.Rg5 Rh6 and the pin disables 10.Sc4) 8.Rg5 Rh6 9.Qd3 Qa5+ 10.Sd2 Qa8 11.Sc4 Rh4. The second solution also begins symmetrically: 1.d3 d6 2.Be3 Be6 3.Bxa7 Bxa2 (similarly not 1.d3 d5? when the black bishop is blocked from both a2 and c2) 4.Bb6 Bb3. Here the opened 4th and 5th ranks save both rooks a move, which compensates for the extra steps used by the d-pawns. 5.Ra5 cxb6 6.Rg5 Ra4 7.cxb3 Rh4 8.d4 b5. Again the potential queen check on a5 forces the move order; not 8…d5? 9.Qd3 b5 10.Sd2 Qa5 11.Sc4?? 9.Qd3 Qa5+ 10.Sd2 Qa8 11.Sc4 d5. Three pairs of white and black units – the bishops, rooks, and d-pawns – alter their routes across the two phases, while the identities of the b3- and b5-pawns are also changed.

Andy Sag: It’s clear that each queen’s bishop is captured by either the a- or c-pawn on b6 and b3 respectively, with each side left with two spare moves. The black queen check from a5 ensures that the white knight must go via d2.
Jacob Hoover: This was moderately challenging, but I’ve never found any proof game problem to be no-brainer-level easy.

As long as the white queen keeps protecting its two officers, Black cannot move either the bishop or the rook without allowing an immediate mate. Black is therefore almost in zugzwang – …Sg6 permits Kxg6 – but the strong defence …Sf7 must be catered for by White. The try 1.Qh2? threatens 2.Qxb2/Bxb2 (while preserving the guard of e5), but no progress is made after 1…a3 2.Qe2 or 2.Rh1 Sf7! The key 1.Qd4! (2.Qxb2) works by observing the d-file, in anticipation of the knight’s play from the f7-square. After 1…a3 2.Rd1 Sf7 3.Bf6, White has encircled the knight, which due to zugzwang can no longer escape capture. 3…Sd8 4.Qxd8, 3…Sd6 4.Qxd6, 3…Se5 4.Bxe5, 3…Sh8 4.Bxh8, 3…Sg5 4.Kxg5, 3…Sh6 4.Kxh6. For instance, after 3…Sd8 4.Qxd8, 4…Rc1 5.Rxc1, 4…Rxd1 5.Qxd1, 4…Bc3 5.Bxc3, 4…Bd4 5.Bxd4/Qxd4, 4…Be5 5.Bxe5, and 4…Bxf6 5.Qxf6. The whole white force coordinates to trap the black knight, while carefully maintaining control of the two pinned pieces.

Andy Sag: After using a mate threat to lock up the a1 corner, White sets up a clever ambush on the knight. Noted a few minor Q/B duals on the mating move which appear to be unavoidable.
Jacob Hoover: This is the Grab theme.
Bob Meadley: Well, I may have got it quickly; it’s a question of restriction. Looks like Romeo has a way to go to catch me.

All possible black moves in the diagram are provided with set mates. The black rook demonstrates focal play in 1…R~rank 2.Qa8 and 1…R~file 2.Bxd3; 1…b3 also permits 2.Bxd3. The g2-knight shows correction play with 1…S~ 2.Qe1, 1…Sxf4 (creating a flight on d5, but self-pinning) 2.Qh1, and 1…Se3 2.Sd6. Lastly, 1…exf4 enables 2.Qd4. White has no way of maintaining the block position, however, e.g. 1.Rf1? Sxf4!, 1.Kf6? Ra6+! The queen has many strong tries that generate one or more threats, such as 1.Qa2? (2.Qc4/Qd5/Sxg2) exf4 2.Qe6 (changed mate), but 1…Se3!, 1.Qd1? (2.Qf3) exf4!, and 1.Qh1? (2.Qxg2) exf4! The surprising key 1.Bb3! unpins the black knight and threatens 2.Bd5. Much of the set play is retained when Black counters the threat. 1…Rxb3 2.Qa8, 1…Sgxf4 2.Qh1, 1…Se3 2.Sd6, and 1…exf4 2.Qd4. The unpinned knight produces a fine new variation, 1…Sdxf4 2.Qb1, which nicely parallels the other knight’s self-pin on the same square. This terrific example of the block-threat form is very deceptive, because the solver is reluctant to release the black knight and abandon the set variations, 1…R~file 2.Bxd3 and 1…Sg~ 2.Qe1.

Andy Sag: The key unpins the d3-knight adding a diagonal queen pin-mate.
George Meldrum: Very enjoyable problem to solve.
Bob Meadley: A mesmerising two-mover.

A queen mate on f7 would cover the g7-flight, but clearing the b3-f7 diagonal is not straightforward. For the set play, an en passant capture of the c5-pawn is permitted only if we could prove by retro-analysis that Black’s last move was c7-c5. Most of Black’s units are enclosed and couldn’t have just played. The previous move wasn’t Kf6-g6 or Kf5-g6 as each would result in an impossible check delivered by the g8-knight or e4-pawn (neither has a legal retraction). Thus only the c5-pawn could have just moved, and it wasn’t c6-c5 as that would mean another illegal check. So Black’s last move was indeed c7-c5 and the set play is 1…dxc6 e.p. 2.cxd3 Qxf7. In most helpmates with a thematic set play, it can’t be repeated in the actual solution due to the lack of a black tempo move. Here Black has plenty of waiting moves available, like 1.exd6, but the set sequence can’t follow because of the peculiar en passant rule. A different way to guard the g7-flight needs to be organised, and the job falls on the e4-pawn, which is also able to open a mating line for the bishop simultaneously, by means of another special pawn capture. 1.cxb3 e5+ 2.f5 exf6 e.p. Therefore en passant play occurs twice in this retro-helpmate hybrid, first as an initial move and then as a mating move. The Zilahi theme is expertly added to the mix, in that the white queen and bishop exchange roles in giving mate and getting sacrificed.

Andy Sag: Set play is a retro as Black’s last move had to be c5 (from c7). En passant rules OK!
George Meldrum: The set play was a great help in finding the solution.
Jacob Hoover: Both phases of play involve en passant. Interesting.

White’s knight is able to unblock either pawn on the second rank and then control the e7-flight, but neither pawn could mate on d5 or f5 as these squares are too well defended. Still, releasing a white pawn is a likely start, especially since this involves the thematic capture of its pinner. 1.Sd2 2.Sxb1 3.Sxc3. Now we expect the knight to move only once before the pawn trek begins, otherwise the move sequence won’t be precise. With six more moves available, the c2-pawn can promote and then a queen mate on d6 will cover the e7-flight. 4.Sb5 5.c4 6.c5 7.cxb6 8.b7 9.b8=Q 10.Qxd6. The second solution commences with White removing the prospectively pinning rook, 1.Sxe1 2.Sg2 3.Sxe3. Regardless of where the knight goes, an analogous promotion of the e2-pawn is not achievable in time. Another plan is required, one that employs the pawn to guard e7 and activate the g4-pawn as well. 4.Sf5 5.e3 6.exf4 7.fxg5 8.gxf6 9.g5 10.Sg7. The two parts begin with matching play but finish on a disparate note, which makes this uncommon series-mate example even harder to unravel.

Andy Sag: The two solutions are quite different but both involve early removal of a pinning piece and a 3rd-rank pawn so a 2nd-rank pawn can do its job.
George Meldrum: The task has a feeling of being impossible given the heavy defence surrounding the king.

Since the rook and bishop are insufficient to set up a mate, the black king must approach some of the pawns to utilise them. In the first solution, the king aims for a4 to make the b5-pawn a block, but surprisingly the a3-pawn needs to be removed to enable a rook mate. Meanwhile, White executes an Indian manoeuvre on the third rank – critical move across e3 followed by a self-interference on that square – to facilitate the king’s trip. 1.Kc2 Rf3 2.Kb2 Be3 3.Kxa3 Bc5+ 4.Ka4 Ra3. Despite the symmetrical position, an equivalent sequence on the other side of the board fails. Instead, the king heads to e5, to prepare for a bishop mate on the long diagonal. Another interference on e3 occurs, though the rook is cut off in the opposite direction. 1.Ke2 Be3 2.Kf3 Bh6+ 3.Ke4 Rd4+ 4.Ke5 Bg7. The bishop’s need to travel all the way to h6 ensures that a corresponding solution on the queen-side doesn’t work. An attractive doubling of the asymmetry theme with two unmissable “tries.” 

Andy Sag: The pattern is symmetrical but the board is not, so the two solutions differ. However, in both solutions the black king makes four moves while the white king does nothing. In each case the rook and bishop must find the correct sequence and tempo, making the problem hard to solve.
Jacob Hoover: Both solutions feature battery creation and a battery play.
George Meldrum: Surprisingly, the only symmetry in the solutions is the move Be3 which allows passage for the black king.
Bob Meadley: Worthy of a high prize. Bloody thing took me too long. Very nice how the loss of the file on the LHS allows the mate but not on the RHS.

In the three-move version of the Lincoln theme (see WP717 above), White plays the same second move against multiple defences, but the ensuing mates are different. Here the fine withdrawal key 1.Kc2! concedes a flight on c4 and threatens 2.b4+ [A] Kc4 3.Sd6 – the threat is thematic because 2.b4+ is the recurring second move seen in the main variations. 1…Rd8 [a] 2.b4+ [A] Kc4 3.Rxc6, 1…Bf8 [b] 2.b4+ [A] Kc4 3.Sxe5, 1…Sxf7 [c] 2.b4+ [A] Kc4 3.Bxf7 (2…Kd6 3.Rd7), and 1…Rg6 [d] 2.b4+ [A] Kc4 3.Be2. Furthermore, if Black takes the flight with 1…Kc4, White plays 2.b4 [A] yet again to threaten 3.Sd6. This creates an insert two-mover in which the b4 move acts like a key that provokes many variations with distinct mates: 2…Rd8 [a] 3.Rxc6, 2…Bf8 [b] 3.Sxe5, 2…Sxf7 [c] 3.Bxf7, and 2…Rg6 [d] 3.Be2. The four black moves are labelled [a/b/c/d] to highlight how they change their functions from first- to second-move defences, and these moves also lead to a kind of mate transference when compared with 2…Kc4. Overall, a clear and intensive demonstration of the Lincoln theme with almost no by-play.

Composer: A new twist to the Lincoln theme where black defenses show up as first and second moves, all resulting in transferred mates. This treatment of the theme (which isn’t easy to implement) probably has to be called the Lincoln-2 theme.
Andy Sag: The royal key gives a flight but unleashes the deadly 2.b4 which is played in all variations.
Jacob Hoover: None of the defenses dealt with the pawn check. This was quite the interesting problem.
George Meldrum: The innocuous pawn on b2 is put to use with 1.Kc3?, and the problem responds brutally with “don’t even think about it” [1…exf4+!].  A large array of tricky variations ensues after 1.Kc2!, and the black king is never mated on its original c5-square.

Only three moves are available to Black, and White has three thematic tries (all waiting moves) that handle two of them but fail to a third. 1.Kf8? Kh4 [a] 2.Sf3, 1…Kh6 [b] 2.Sf7, but 1…fxg5! [c]; 1.h4? Kh6 [b] 2.Qh7, 1…fxg5 [c] 2.Qxg5, but 1…Kxh4! [a]; and 1.h3? fxg5 [c] 2.Qh7, 1…Kh4 [a] 2.Qg4, but 1…Kh6! [b]. The three black moves thus rotate their functions as normal defences and unique refutations, giving rise to the cyclic refutation theme. Remarkably, in these try-phases every defence leads to a pair of changed mates. The key 1.Kf7! (waiting) provides for all three defences: 1…Kh4 [a] 2.Sf3, 1…Kh6 [b] 2.Qg6, and 1…fxg5 [c] 2.Qh3. Yet two more new mates are induced, with only 1…Kh4 2.Sf3 repeated from the virtual play. In an “ideal” presentation of cyclic refutation, three changed mates occur against each of three defences to bring about nine distinct variations, so the eight shown here is nearly perfect. That this is all accomplished with just six pieces makes this one of the greatest two-move miniatures.

Andy Sag: Quite a busy miniature waiter. Three variations and three tries with only six pieces!
Jacob Hoover: Notice that there is a cyclic pattern here. Niiiiiiiiiiiiiiiiiiice.

White wants to threaten a queen check on f7, to force 2…Bxf7, but this must be done without controlling Black’s R + B and B + S batteries (e.g. the non-thematic 1.Qd5? has multiple refutations like 1…Bd1+ and 1…Sa6+). Both the thematic try and key unpin the d7-pawn, which defends by creating a flight on d7. After 1.Qf3?, the two pawn moves provoke different responses, based on the queen’s ability/inability to return to c6. 1…d6 [a] 2.Qc6+ [A] (switchback) Sxc6, and 1…d5 [b] (interference) 2.c8=Q+ [B] Rxc8. But Black thwarts the try with 1…hxg6! After the key 1.Qf6!, the queen’s access to c6 from a new direction brings a change of continuations, against the same d7-pawn defences. 1…d6 [a] (interference) 2.c8=Q+ [B] Rxc8, and 1…d5 [b] 2.Qc6+ [A] (switchback) Sxc6. If 1…hxg6 then 2.Qxg6+ Bf7. The reciprocal change of white moves (from AB to BA) is rendered with a neat mechanism that has a pleasant geometrical aspect.

Andy Sag: Black has some tricky “defences” but White has the answers. First prize well deserved.
Jacob Hoover: Themes: unpins, reciprocal change.

White’s knight on b3 has two potential mating moves to c5 and d2, each of which is guarded by the e7-bishop or h2-rook. The imprisoned white king can’t be used to observe the f5-flight, so Black must block it, though all immediate attempts to do so would check (or mate) White. The twinning, which shifts the f3-pawn, provides a hint that Sd2 will occur in part (b) to recover the f3-flight, so Sc5 is the likely goal in (a). The thematic try 1.Bc5? unguards the mating square and also dismantles the B + P battery aimed at the white king, such that the pawn could self-block on f5. However, after 1.Bc5? White lacks a waiting move, because the bishop is needed on a2 to control d5 while 1…Rxa4 would be a disruptive check. Therefore Black begins with 1.Bb4 (still defending c5), in anticipation of 1…Rxa4 which pins the bishop and permits 2.f5 Sc5. Part (b) presents analogous tempo effects. The try 1.Rd2? unguards d2 and removes the R + B battery on the h-file, allowing the bishop to self-block on f5. But after 1.Rd2? White has no waiting move, as the a3-rook needs to control d3 while 1…Bb1 would check. So Black plays 1.Rc2 (retaining guard of d2), preparing for 1…Bb1 which pins the rook and facilitates 2.Bf5 Sd2. White’s 1…Rxa4 and 1…Bb1 in the respective solutions are not tempo moves as such, because they serve an essential pinning function, but the logic of the play reveals unmistakable tempo strategies.

Andy Sag: The twin has matching themes. First black move deactivates an unwanted battery then White pins that piece. Second black move can then block f5 without checking. White can now use the b3-knight to deliver pin-mate.
George Meldrum: The b3-knight is set to mate, if the f5-square is blocked off, and either the c5- or d2-square is unguarded, and White can make an intermediary waiting move. How Feather pulls this off is quite amazing.
Rauf Aliovsadzade: Always a pleasure to analyse this great helpmate composer’s works!

Black has only three possible moves, including 1…exf3 answerable by a short set mate 2.Sxe6, but providing for 1…c5+ and 1…e3 proves difficult. 1.Kc5? still fails to 1…e3! while 1.fxe4? doesn’t help against 1…c5+! 2.Sxc5 Kxe5. The unlikely looking 1.Sc5! (waiting) gives up the rook and prevents the checking defence directly, while arranging 1…e3/exf3 2.Sd3 (the latter changes the set mate). After 1…Kxe5, the black king has no less than six flights, two of which are removed by 2.fxe4. The four diagonal moves left elicit different mating responses, to produce the star-flights theme. 2…Kd4 3.Sf3, 2…Kf4 3.Sd3, 2…Kd6 3.Sf7, and 2…Kf6 3.Sd7. The excellent key not only sacrifices the rook but grants a flight on e5, from which the king initiates the thematic play.

Andy Sag: The key must deal with an unprovided check so there are not many moves to consider. It does, however, set up nice star-flights if the rook is accepted (but allows a short mate if not, due to d3 becoming unguarded).
Jacob Hoover: The key nullifies the strong 1…c5+ defense. Theme: symmetry (kind of), black king-star.
Bob Meadley: Loydish in looks with those columns of pawns. George is generous giving away that castle.

Numerous black checks are possible in the diagram, all with mating replies prepared. The key 1.d8=S! threatens 2.Sc6 and preserves the set play. The two black knights, front pieces of two batteries aimed at the white king, are half-pinned by the h5-rook. When either knight gives a discovered check, White exploits the pin of the remaining piece: 1…Sf~+ 2.Qxe6 and 1…Sg~+ 2.Qxg7. Each knight can correct its self-pinning error with a specific placement, but this inadvertently unpins the white knight: 1…Sd4+ 2.Sxe6 and 1…Se4+ 2.Sg6, so White delivers two battery mates against the battery defences. There’s by-play with 1…Rxf4+ 2.Bxf4 and 1…Bd5/Bd7 2.Qd5. Plenty of strategic effects are packed in the four main variations – battery play, half-pin, unpin of White, along with black correction play.

Andy Sag: The threat of mate on c6 by a promoted knight is impossible to defend without exposing fatal weaknesses. Note 1…Rxc3 does not prevent the threat.
George Meldrum: Nice array of set play in the provided checks to the white king.
Jacob Hoover: Themes: Underpromotion, black correction, battery plays.
Paz Einat: A beautiful half-pin cross-check combination. The composer probably had a hard time finding a good key and the one he used is a decent compromise.

Two potential mating moves in the diagram are Sxc7 and Bxf7, both of which cover the d5-flight. Black has to promote a pawn in order to self-block the remaining flight on e7, and this requires capturing a white piece that’s superfluous when the other one mates. 1.c6 2.cxb5 3.b4 4.b3 5.b2. A queen or rook on e7 would prevent the mate, so 6.b1=S 7.Sc3 8.Sd5 9.Se7 for Bxf7. 1.fxg6 2.g5 3.g4 4.g3 5.g2. Here a promoted knight would take one step too many to reach e7, hence 6.g1=B 7.Bf2 8.Bh4 9.Be7 for Sxc7. Since White’s knight and bishop exchange their functions of getting sacrificed and giving mate, the Zilahi theme is shown. Each of these white pieces also performs a mutual capture with a black pawn (from c7 and f7 respectively) across the two solutions. Furthermore, Black’s choice of promoted piece in each case matches the piece just captured by the pawn. All of these connecting features, capped off by the ideal-mate finishes, more than compensate for the easy solving.

Andy Sag: Curiously, the pawn that promotes is the one that gets captured by the mating piece in the other solution. Solved this without setting it up on a board.
Jacob Hoover: We have the Zilahi theme, and also a doubling of the Excelsior theme (since each promoting pawn moved all the way from its starting square to promote). Additionally, the black c- and f-pawns swap roles between promoting and getting captured.
Thomas Thannheiser: Zilahi with underpromotion to bishop or knight after capturing this piece. Very nice!

Mating the black king on g3 proves infeasible, partly because of the obstructing white pawns. A plausible final square for the king is f3 where many nearby pawns work as blocks or guards; indeed, if the e4-bishop were protected and g3 were blocked, the king would be mated there. This scheme requires the bishop to execute a switchback, and the piece seems to have multiple options as the first move. However, eliminating such a dual turns out to be the theme of the problem, as the bishop must carefully avoid interfering with the rook and the black queen too when the latter goes to g3. 1.Qg7 Bh7 (not 1…Bg6/Bf5?) 2.Kxf3 Re5 3.Qg3 Be4. Here the alternative queen route of Qb8-g3 fails because it’s intercepted by 2…Re5. For the analogous second solution, the king could be mated on h4 by the h5-rook, assuming the white piece is guarded and g3 is blocked. Similarly, the rook must make a switchback and it has three apparent ways to do so, but two of them would cut off the bishop or the black queen. 1.Qb8 Rg5+ (not 1…Re5/Rf5?) 2.Kxh4 Bg6 3.Qg3 Rh5. This time the queen cannot choose the Qg7-g3 route as it’s closed by 2…Bg6. Intriguing dual avoidance by the two white pieces, which effectively sidestep a Grimshaw interference on f5.

Andy Sag: One good switchback deserves another! White rook and bishop exchange switchback and guarding roles. In each case the queen must block g3 but using different routes.
Satanick Mukhuty: Impressive switchbacks and ODT. I really enjoyed how the black queen’s path to g3 separate the two solutions.

White’s limited force has to generate a threat and deal with the unprovided flight on d5. The key 1.Qd2! removes that flight but grants a new one on e5, and threatens 2.Rf4+ Ke5 3.Qd6. Two defences are distant self-blocks that are answered by different knight checks and rook mates: 1…Rxf6 2.Sc5+ Ke5 3.Re2 and 1…Se6 2.Sd6+ Ke5 3.Rxf5. Another distant self-block on e6, 1…Re6, enables 2.Rf4+ Ke5 3.Qd4. 1…d5 is a standard self-block that’s exploited by 2.Qf4+ Kd3 3.Rd2. Lastly, the flight-move 1…Ke5 permits 2.Re2+ Kxf6 3.Qh6. Good coordination of the three white pieces brings about six varied mates including the threat, with black self-blocks as the common motif in four variations.

Andy Sag: Unprovided flight Kd5 limits the number of potential keys but the give-and-take key is still quite hard to find as expected from Leonid. I really like the surprising Qh6 variation.
George Meldrum: Good key-move and fantastic variations – I especially like the one where the white queen mates on h6.

In the initial position, the white queen is well-placed to handle 1…B~ (gate-opening) with 2.Qxe7 and 1…Re8 (unguard) with 2.Qxd7. The thematic try 1.Qc2? (waiting) discards these set variations and concedes a flight on d6. Then either 1…B~ or 1…R~ opens a white line of guard to d6 and enables 2.Qf5, a changed mate for each set defence, while 1…Kxd6 leads to a pin-mate 2.Qc6. This try is, however, refuted by 1…f5! The key 1.Qc1! (waiting) likewise withdraws the queen and abandons the set play. Now the line-opening weaknesses of 1…B~ and 1…R~ are exploited by 2.Qe3, yet another changed mate. Taking the flight with 1…Kxd6 allows 2.Qc6 as seen previously. Lastly, 1…f5 2.Qh6 shows the queen regaining control of the lateral flight. Fine changed play across three phases for 1…B~ and 1…Re8 (though the repeated queen mates against these defences prevent a Zagoruiko realisation), complemented by a splendid sacrificial key.

Composer: In this Meredith setting the key grants a flight-capture answered by a pin-mate, one of three widely separated orthogonal queen mates. Another try is 1.Qc3? f5!
Rauf Aliovsadzade: Nice key!
Bob Meadley: The white queen was on her morning stroll.
George Meldrum: Neat enough but the black pawn move to f5 gives a flight to the black king, and so is a real giveaway to finding the solution. Notable is the move 1.Bxe7, threatening 2.Qxd7 – it’s not stopped by 1…Kxe7 2.Re8 [a different pin-mate!]. However, black bishop moves thwarts this attempt.

White cannot organise an ordinary back-rank mate like …Qg8 mainly because of the a7-knight, while arranging …Qxb7 also proves infeasible, e.g. 1.Bc5 Qf3 2.S5c6? The correct goals are to use the queen and the rook to mate on b8, with each piece supported by the other. This scheme means shifting the white king off the queen’s diagonal and both black bishops off the b-file. In one solution, the b6-bishop cuts off the h8-rook and also gives the white king access to e3. The resulting king move to the third rank, however, self-pins the two prospective mating pieces; Black is thus forced to allocate the b7-bishop’s move to unpin the queen by interposition. 1.Bd8 Ke3 2.Bf3 Qb8. In the other part, the b7-bishop takes over the tasks of closing the eighth rank and providing a king’s flight, now to f3. The king’s move again self-pins the two white pieces, so the b6-bishop must interpose to unpin the rook. 1.Bc8 Kf3 2.Be3 Rb8. The two matching solutions involve exchange of functions between the two black bishops and between White’s queen and rook. Curiously White and Black alternate in playing to the same squares, e3 and f3.

Andy Sag: In each case a black bishop blocks the h8-rook laterally and unguards a square on rank three; the white king occupies the unguarded square, self-pinning both pieces. The other bishop unpins one of the pieces which then mates on b8.
George Meldrum: The black bishops reverse their roles in each providing a pleasant pair of solutions.

The thematic key 1.Rcc8!, with a short threat of 2.Rcg8, uncovers the black B + R battery aimed at the white king. Black can defend only by firing the battery, giving check with four different moves of the rook. One pair of rook moves interferes with the g2-bishop along the long diagonal, allowing the white king to hide on b7 or c6. Each rook defence, however, contains a new checking resource that forces the king to pick one particular square. 1…Re4+ 2.Kb7 and 3.Rcg8 (not 2.Kc6? Rc4+!). 1…Rf3+ 2.Kc6 and 3.Rcg8 (not 2.Kb7? Rxf7+!). The other pair of rook moves prospectively cuts off the g2-bishop when …Bf1 is played, enabling the white king to safely land on a6 or b5. Again each rook defence compensates with a new potential check that determines the correct choice for the king. 1…Rd3+ 2.Ka6 and 3.Rcg8 (not 2.Kb5? Rxd5+!). 1…Re2+ 2.Kb5 and 3.Rcg8 (not 2.Ka6? Ra2+!). A black rook cross is answered by a white king cross, with White even imitating the direction of every black step. The double theme seems appropriate for the Christmas season!

Andy Sag: The key threatens a short mate but grants four battery checks using the black rook as a trigger, each answered by a copy-cat pattern of moves by the white king leaving black defenceless against the original threat.
George Meldrum: The key leaves the white king exposed, without any clothes. Kind of cute how the white king replicates the black rook’s moves to avoid further checks.
Rauf Aliovsadzade: Great problem. A classic in my view.

If we assume this is a standard directmate problem in which White begins and forces mate in two moves, there is no solution. However, the position is illegal if it is White’s turn, since neither of the two black units could have made the last move – the king did not come from g2/h2 where it would be adjacent to the other king, and the pawn is on its starting square. By convention, in such a situation we may infer that White made the previous move (which preserves legality), and so it is Black’s turn to play. Black’s four possible moves lead to various positions in which White can mate in two, each solved by a different key. If …Kxg1, then 1.Rf6! Kh1 2.Rf1. If …fxe6, then 1.Sh3! e5 2.Be4. If …f6, then 1.Sf3! f5 2.Rh6. If …f5, then 1.Kf2! Kh2/f4 2.Rh6. This tricky retro two-mover incorporates a remarkable number of distinct “variations” for the type, especially given the miniature setting.

Andy Sag: Set position is illegal with White to move as there is no way Black could have moved into this position. Therefore it must be a Black-to-move problem.
Thomas Thannheiser: Black to move!
Bob Meadley: Christmas Trickster. Nice!
Satanick Mukhuty: Clever indeed!
George Meldrum: Had me going on this one. Neat solution for such a light setting. Good end-of-year problem.