Paradox

One of the most intriguing aspects of chess composition is in its depiction of the paradoxical. In themes of this variety, the strategy involved seems to defy the normal principles of play, making the solution particularly surprising and attractive. Basic paradoxical ideas include the sacrifice, and other kinds of apparently self-weakening moves that prove to be advantageous. In the problem context, however, the term paradox usually refers to special instances of counter-intuitive play that are more refined and rarely, if ever, achievable in a practical game.

In Problem 121, we note that after 1.Rf5+? Ke6! or 1.Sf4+? Ke5!, White can make no further progress. The key 1.Bc5! threatens 2.Rxd6+ exd6 – opening the e-file for the white rook – 3.Sf4. The two principal defences are made by the e7-pawn: 1…e6 attacks f5 but it is met by 2.Rf5+ exf5 (or 2…e5) 3.Sf4, and 1…e5 attacks f4 but White answers with 2.Sf4+ exf4 3.Rf5. The paradoxical scheme here is that White plays to a square only after it has been guarded by Black (an effect made possible by the black pawn’s self-blocks on e6 and e5). These two variations incidentally also display a formal connection, in that White makes the same two moves but in reverse order. There is by-play: 1…Bh6 2.Rxe7 (threats: 3.Rxd6 and 3.Rf5) dxc5 3.Rd7, 2…Bf4 3.Sxf4. 1…dxc5 2.Rxd8. 1…exf6 2.Sf4.

Many problem themes are based on how certain moves change their functions when they recur in different parts of the solution. Thus in the previous three-mover, White’s moves Rf5 and Sf4 occur in both main lines, but their functions as a second move continuation and a mating move are interchanged. Sometimes the function change is paradoxical in nature because the different purposes of the same move seem to contradict each other. The Dombrovskis theme presents such a situation. Named after a Latvian composer who originated the concept in the 1950s, it is illustrated in the next two examples.

Problem 122 involves two white knight tries, and the thematic moves consist of the white mates threatened by these tries and the black defences that refute them. The first try 1.Sd6? threatens 2.fxe4, but because the knight move has interfered with the queen’s control of e5, Black refutes by 1…c3!, which cuts off the bishop from guarding the same square and so creates a flight. The second try 1.Sd4? threatens 2.Bxe6, and now it is the knight that stops the bishop from attacking e5, and Black refutes by 1…Bd6!, which shuts off the queen and once again permits the black king to escape. The solution changes the relationship between these moves from the try phases, and begins with 1.Sa7!, which covers c6 to threaten 2.Qe5. Black defends with 1…c3 (which errs by opening the rank for the b4-rook), and White’s reply 2.fxe4 is the same mate threatened by the 1.Sd6? try and prevented by 1…c3! So the same black move that defeats a threatened mate in the try play actually provokes the same mate in the actual play. This constitutes the Dombrovskis theme, and it is effected again in the variation 1…Bd6 (giving the e8-rook access to e6) 2.Bxe6, since this contrasts with the other try play following 1.Sd4?, when the threat of the same 2.Bxe6 mate is foiled, rather than enabled, by the very same defence 1…Bd6!

The three-mover 123 splendidly combines the main ideas of the two previous works. The white queen will threaten mate on e5 or d5 when either square is covered by another white piece, but 1.Kf5? is defeated by 1…cxd6! (2.Sf6 Sc7!), and 1.Sf6? is answered by 1…c6! (2.Kf5 Bxd6!). The key 1.Sf8! carries a threat, 2.Qf5 then 3.Se6, which is parried by any move of the c7-pawn, since it would allow 2…Sc7 to defend e6. Now 1…cxd6 facilitates 2.Qxe5+ in spite of the added defence of e5, 2…Kxe5 3.Bg7 – exploiting the self-block on d6 – and 2…dxe5 3.Se6. In addition to the “twist” of White sacrificing on a further protected square, the Dombrovskis theme is seen, since 1…cxd6! has previously disabled the threat of 2.Qxe5+ in the try play of 1.Kf5? The second thematic defence 1…c6 generates similar play: it observes d5 and yet provokes 2.Qxd5+, due to the self-block on c6 after 2…Kxd5 3.Rd1, while 2…cxd5 allows 3.Se6. And the Dombrovskis effect is realised in that 1…c6! has functioned as the refutation against the threat of 2.Qxd5 in the 1.Sf6? try. (A minor variation is 1…S5-any 2.Rc4+ dxc4 3.Qxc4.)

In Problem 124, neither 1.Bxd8? nor 1.Bxf8? (threat: 2.Qe7) will work immediately because of 1…Bh4! The key 1.Sg5! closes the h4-e7 diagonal to rule out that bishop defence, but White still does not threaten 2.Bxd8 or 2.Bxf8 because the key has also opened the e-file, permitting 2…Qxe5! Instead, White’s threat is the square-vacating sacrifice 2.Qe6+ and then 3.Sf6 regardless of Black’s reply (capture of queen or 2…Ke8). Black’s thematic defences, 1…Sde6 and 1…Sfe6, are paradoxical because they place a piece on the very square that White is threatening to move to, and still thwart that threat: 2.Qxe6+? would allow the remaining black knight to recapture on e6 with a discovered check. The correct response to 1…Sde6 is 2.Bd8, another paradoxical move that puts a white piece on a square only after it has been vacated (and guarded) by a black piece. Now 3.Qe7 is unstoppable, because the knight defence has closed the e-file and rendered 2…Qxe5 ineffective (also note that 2.Bxf8? fails to 2…Rxf8+!). Likewise, 1…Sfe6 prompts White to occupy the newly vacated square with 2.Bf8, followed by 3.Qe7. A remarkable solution in which the pieces seem to follow one another in various chains.

Since helpmates require the cooperation between the two players, paradox in this genre generally arises when Black’s play seems to hinder White. A popular idea in this vein is the capture of white force. Many examples of such sacrificial play have been given in previous articles, however, so here we quote Problem 125, a delightful helpmate showing another counter-intuitive idea. White plans to mate with the rook, but ironically Black has to pin it first, 1.Qa1 Rb1 2.Bd1 Rxb7. The twin (b) has the white king starting on b1, and this leads to another sequence of pin and unpin, 1.Rh1 Rg1 2.Sd1 Rxg2.