Economy of force is one of the basic principles in chess composition – it means that a problem should use the minimum amount of material to bring about a given idea. An interesting way to express this type of economy is to limit a player (usually White) to employing just one piece besides the king. Problems that meet this criterion are called minimals.

A comprehensive anthology of the “best” minimal problems, Minimalkunst im Schach, was published in 2006, though I acquired a copy only recently. This impeccably produced hardback contains more than a thousand problems, arranged in these genres: studies, directmates, helpmates, and fairies. Each section is subdivided according to the type of piece owned by the minimal side, next to the king. Brief comments in German accompany the solutions, but you don’t need to understand the text to enjoy these well-selected compositions.

The process of compiling this album was a mammoth task that began in 1980, as co-author Hilmar Ebert relates in the Introduction (also in English). He worked through a preliminary selection of some 30,000 problems and picked 10,000 of them to present to his fellow editors, Hans-Peter Reich and Jorg Kuhlmann. Over the years the trio meticulously analysed and assessed these minimals for their quality, originality and correctness, and eventually chose 1032 for inclusion in this book.

Here I quote two representative problems from the collection, both marvellous works. The first features an unusual helpmate idea in which White’s initial play is motivated by the need to stop an imminent black mate. Due to zugzwang, Black is about to give mate in three moves at the latest, e.g. 1.h1(S) Sc6 2.h2 Sb4 and now 3.Sg3/e2 mate cuts short White’s plan to play 3…Sd3. To avoid such a hazard, Black must promote carefully: 1.h1(B) Se6 2.h2 Sf4 3.Bg2+ Sxg2. Now 4.h1(S) fails because after 4…Sxe1/Sf4, Black is again forced to mate with any legal move. So instead 4.h1(B) Sxe1 (self-pin) 5.e2+ Kxf2 (unpin) 6.Bg2 (tempo move) Sd3.

A helpstalemate problem is similar to a helpmate except that the players cooperate to put Black in stalemate. In Aussie composer Ian Shanahan’s example, we see an elegant delivery of the Allumwandlung theme. 1.f1(S) Kc6 2.Sg3 hxg3 3.c1(B) g4 4.Bh6 g5 5.h2 gxh6 6.h1(R) hxg7 7.Rh8 gxh8(Q) 8.Ka4 Qc3. The four types of promotions occur in ascending order; moreover, this miniature incorporates an Excelsior and an ideal stalemate.

In a so-called “massacre” proof game, just a handful of pieces remain in the diagram position, and the preceding play largely consists of captures by both sides. The nature of such games, where virtually no clues of what transpired are left in the final position, means they require computer assistance to be constructed (or solved). There’s a sort of “Holy Grail” for problemists to find a massacre game with a unique solution that leaves only the two kings on the board. Earlier research by programmers had proved that no such games exist in 16½ or 17 moves (except those with additional problem conditions). However, by extending the length of the games to be analysed and limiting the search based on how far the kings will travel, Francois Labelle of Canada has discovered the first two-king SPG that is completely orthodox.

This astounding achievement took a staggering amount of computer time. As Labelle reports in an article published in StrateGems last year, ten months were spent on the searching parameters that yielded the successful game in 19½ moves. The problem position, shown on the right, is quite nice to boot – a symmetrical set-up with both kings on their original file. The solution is 1.c4 e5 2.Qb3 Qh4 3.Qxb7 Qxh2 4.Qxb8 Qxg1 5.Rxh7 Rxb8 6.Rxg7 Rxb2 7.Rxf7 Rxa2 8.Rxd7 Rxd2 9.Rxa7 Kxd7 10.Rxc7+ Kd6 11.Rxc8 Qxg2 12.Rxf8 Kc5 13.Rxg8 Rxg8 14.Bxg2 Rxg2 15.Sc3 Rxf2 16.Kxf2 Kxc4 17.Kf3 Kxc3 18.Bxd2+ Kxd2 19.Ke4 Kxe2 20.Kxe5.

Labelle’s method also generated 349 sound SPGs with three pieces left, some confirming previously known results, but plenty of which are new works. Here’s an interesting one that shows the black king trekking to the farthest possible square. 1.d4 h6 2.Bxh6 c5 3.Bxg7 Rxh2 4.Bxf8 Rxh1 5.Bxe7 Kxe7 6.Sc3 Kd6 7.dxc5+ Kxc5 8.Qxd7 Rxg1 9.Qxb7 Rxg2 10.Qxa7+ Kb4 11.Qxf7 Rxf2 12.Qxg8 Rxe2+ 13.Bxe2 Rxa2 14.Bd3 Qxd3 15.Qxc8 Rxb2 16.Qxb8+ Kxc3 17.Qxb2+ Kxb2 18.cxd3 Kxa1. While it doesn’t sound very likely, I wonder could any of these three-unit problems be “twinned” with one another!?

Are there more two-king SPGs to be discovered? It’s possible, according to Labelle, whose analysis isn’t exhaustive (though he has conducted a full search of 18-move games and determined that the task cannot be accomplished at that length). Out of more than 3500 possible two-king positions, he has verified the SPGs for

about half, from which one sound setting emerged. Based on this and the data from the three-unit problems, Labelle estimates the chance of another two-king position having a unique solution to be about 1/1000. “Finding it,” he writes, “will not be easy and I fear it will require more computing power, a better pruning method, or plain luck.”

The ‘Problem Potpourri’ column in Australasian Chess continues to draw splendid works from around the world. It’s not easy to pick the highlights from so many fine originals, but here are two of the best that appeared in the year 2012.

The three-mover is by regular contributor Leonid Makaronez, an IM of composition from Israel. The subtle key 1.Bf7! contains a quiet threat, 2.Rf3 followed by 3.Bb2 (which answers 2…Ke5 too). Black’s thematic defences are distant self-blocks: 1…Sb3 2.Re4+ Kc3 3.Rc4, and 1…Rxf6 2.Rd3+ Ke5 3.Rd5. There’s lovely by-play with 1…Sc6 (which stops the threat in view of 2.Rf3? Ke5!) allowing 2.Bd2 and 3.Re4, because 2…c1(Q) no longer checks. Also note that 1.Bg8? is refuted by 1…Rg6!

The helpmate is by another IM, Christer Jonsson of Sweden. The black pieces seem already well-placed to facilitate a knight mate on c2 or b3, but if White plans 1…b5 2…Sb4 3…Sc2 or 1…d3 2…Sd2 3…Sb3, Black is unable to maintain the set-up while using up three moves. Instead, Black helps to clear a path for each white knight by capturing on b4 or d2. White then has a spare move with which to sacrifice a knight to a black pawn, thereby supplying Black with a tempo move. 1.Sxb4 Sg3 2.Sa2 Sb4 3.hxg3! Sc2, and 1.Sxd2 Sf4 2.Sb1 Sd2 3.gxf4! Sb3. Since the two white knights take turns to be sacrificed and to give mate, the Zilahi theme is realised as well, besides the unusual black tempo play provision.

While searching for a suitable Weekly Problem for this site, I found an interesting two-mover by Henry Tate (the Australian problemist who coined the term “fairy chess”), diagrammed below. Although the problem has a fine key 1.Sd4! (threat: 2.Ke5) that sets off good battery play, the ensuing number of thematic variations was disappointing. Black’s knight on e4 is enabled by the unpinning key to give various discovered checks, and its random placement creates a dual: 1…S~+ (e.g. 1…Sf2+) 2.Kd6/Kd7. The correction move 1…Sxf6+ prompts 2.Kxf6, and while 1…Sxc5+/Sd6+ induces one of the dual mates, 2.Kd6, the other one, 2.Kd7, is never uniquely forced. There’s a nice secondary line, 1…Kxd4 2.Sb3.

I wondered if the position was amenable to show more accurate battery variations, and was pleasantly surprised by what could be accomplished. First I added a white pawn on d6 to ensure that the random defence, 1…S~+, is answered by just one mate, 2.Kd7. This means that only after Black has played the correction move, 1…Sxd6+, is White allowed to play 2.Kxd6, making this a distinct variation. However, blocking d6 also makes the problem unsolvable, as 1…Sxc5+! (attacking d7) would refute the intended key. What if we remove the black pawn on g6 as well, so that 1…Sxc5+ leads to 2.Kf5 mate? Without the g6-pawn, the position again involves a dual after the random 1…S~+, viz. 2.Kd7/Kf5. But such a dual is not a serious weakness when these mates also appear individually in other variations, and we find – by good fortune – that the line 1…Sg3+ forcing 2.Kd7 is already in place without further adjustment, to complement 1…Sxc5+ 2.Kf5.

This new version of the two-mover contains four dual-free, royal battery variations – two more than that in the original – and it was used as the Problem of the Week, No.111. (The a7-pawn was also removed as computer-testing confirmed that it was superfluous.) The process of improving this directmate that I’ve described gives an inkling of what constructing a problem involves. If you have been solving problems from this site but have yet to try your hand at composing your own works, I encourage you to do so. Some excellent resources that deal with the techniques of making a chess problem are available online. Composing the Twomover by the Slovakian IM Juraj Lörinc takes you through the course of producing a sample problem step-by-step, starting with a basic theme.

The only book-length treatment of the subject, Adventures in Composition (1944), is by one of the greatest practitioners of the art, the late GM Comins Mansfield. This classic work reveals the thought processes behind the creation of many famous two-movers, and is invaluable to any new composers. You can download a PDF-copy of Adventures in Composition from Vaclav Kotesovec’s site (scroll down to “A. C. White – The Overbrook Series”). Another great resource is the periodical, The Problemist Supplement, which sometimes covers the topic of problem construction directly and is, in any case, a fine problem publication that caters for newcomers. A complete archive of The Problemist Supplement, currently edited by Geoff Foster, is accessible from the site of the British Chess Problem Society.

As a casual but keen reader on the subject of philosophy, I was curious to come across the title, Philosophy Looks At Chess, edited by Benjamin Hale. It is an anthology of articles that explores various aspects of chess from a philosophical point of view. Many topics in which chess and philosophy intersect are examined by twelve professionals of either field. For instance: Do chess-playing computers really understand chess? Is choosing the right chess strategy analogous to how to deal with ethical problems? This essay collection proves to be a mixed bag in terms of quality and accessibility, so I cannot recommend it wholeheartedly. However, it does have some intriguing pieces, and here I will focus on one of its more readable chapters.

That chapter, ‘To Know the Past One Must First Know the Future: Raymond Smullyan and the Mysteries of Retrograde Analysis’, is by Bernd Graefrath. The author is a well-known figure in chess problem circles – he’s the subeditor of a retro section in The Problemist, for example – but I wasn’t aware that he’s also a professor and PhD in philosophy. (On a personal note, some years ago Bernd took the trouble of sending me a nice letter with the news that he, as the judge of a problem tourney that I participated in, had awarded a First Prize to my work!) His subject is Raymond Smullyan, the famous logician and philosopher who published two popular collections of retro chess problems, The Chess Mysteries of Sherlock Holmes, and The Chess Mysteries of the Arabian Knights.

Graefrath discusses some problems by Smullyan and others to introduce readers to the area of retrograde analysis, in which the solver delves into the past of a chess position. He points out interesting parallels between the ideas displayed in these retros and some philosophical concepts, such as “cognitive optimism” and “antiverificationism”. Thus we see how some of Smullyan’s philosophical viewpoints are evoked by his chess compositions! Take Smullyan’s stance against logical positivism, which holds that any statement is meaningless if it’s incapable of verification or refutation. Such a strict verificationism is incorrect, according to Smullyan, and Graefrath quotes a retro problem that is illustrative of this view.

Can White mate in two moves in the diagram position? The answer seems to be ‘yes’, but it’s not possible to demonstrate the key move that would solve it. If Black’s last move was …e7-e5, then 1.dxe6 e.p.! (threat: 2.g8(Q)) would work, and 1…0-0-0 allows 2.Bb7. However, Black’s last move was not necessarily …e7-e5, and so the en passant capture is only a potential key. Alternatively, Black could have made the last move with the king or the rook, in which case it’s illegal to castle now, and 1.Ke6! would solve, followed by 2.g8(Q). But this white king move cannot be confirmed as the key, because Black could indeed have played …e7-e5 last, implying that 1…0-0-0! is still legal as an escape move. Therefore, regardless of Black’s previous move, a white mate-in-two exists in the diagram – yet this “truth” of the position cannot be verified with a particular key move.

The article touches on Smullyan’s fascination with Eastern mysticism, a subject seemingly at odds with the strict logic of his profession. Graefrath writes, “In the final chapter of The Tao Is Silent, Smullyan presents a dialogue in which a metaphysician and a mystic develop a view that may be most attractive for someone with a high regard for logic, mathematics and the sciences, but still thinks that the area of meaningful discourse is not restricted to this area. They may not even cover the most important questions!” Such an attitude happens to coincide with my own, and I share Smullyan’s enthusiasm for mysticism as a sophisticated form of spirituality. I’d recommend The Tao Is Silent as well as another of his books, Who Knows?: A Study of Religious Consciousness, as excellent introductions to mystical thoughts (or non-thoughts!?). Smullyan is a wonderful writer, and for a taste of his style, check out the allegory ‘Planet Without Laughter’, in which a sense of humour, or “getting a joke”, is brilliantly used as a metaphor for spiritual enlightenment.

‘To Know the Past One Must First Know the Future: Raymond Smullyan and the Mysteries of Retrograde Analysis’ is available online as a preview of Philosophy Looks At Chess on Google Books.