Introducing ‘Mutual Mate’ Problems

by Azlan Iqbal
12/31/2021 – Duplex problems are loosely defined as satisfying the same stipulation with the colours reversed, and are typically found in helpmates. In this article, Azlan Iqbal expounds on the much rarer variety that applies to direct mates. Using incidental examples taken from his computer-generated chess problem collection and a couple by human composers, he challenges readers to compose some of their own. For starters you can try to solve the problem shown in our picture. Both White and Black can mate in five, if they have the first move.

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Recently, it occurred to me that some chess problems might be more interesting if the stipulation (e.g., ‘White to Play and Mate in 3’) also held true for the opponent. This would mean that in a position where White was to play and mate in n moves, yet if Black was to play instead from the same position, it would also be mate in exactly n moves. A chess problem that could satisfy both stipulations, i.e., ‘White (or Black) to Play and Mate in N’, would tend to be more difficult to compose and challenging to solve. A ‘two in one’ problem, if you will. We may call these, ‘mutual mate’ problems or, in the case of a study, a ‘mutual win’.

I could not find anything in the existing chess problem literature which already described them. ‘Duplex’ is perhaps an umbrella term that covers them but, to my knowledge, it is not specific to direct mates of the same length for both sides and seems to apply largely to helpmates.

Therefore, I decided to perform an experiment just to see how common such things were ‘in the wild’, so to speak. In fairness, I am confident that if human composers actually tried to compose aesthetically pleasing and thematic direct mate problems of this type, they would succeed as well. Not being much of a composer myself, I turned to my computationally creative computer program, Chesthetica, and its 3,473 compositions from July 2010 to the present (consisting of mates in 2, 3, 4, 5 and studies) which I had been selecting for and compiling. Interested readers can find more information about Chesthetica (pictured below) at the official website.

These 3,473 autonomously computer-generated compositions are the ones I happen to have examined and found aesthetically pleasing, interesting or educational over the years. There are actually presently over 110,000 compositions by Chesthetica; most of which I have never even seen due to the sheer volume. Many more were simply lost as I had discarded the ones I did not see or like for some years prior as well. Only later did it occur to me that it might be prudent to keep them all regardless. The smaller composition set was nonetheless a reasonable sample to test for the new type of chess problem I am describing in this article. While Chesthetica was never explicitly programmed to compose such problems, they might still have occurred. I programmed a subroutine into Chesthetica which would detect mutual mates; however, studies were excluded for now given that mates are decisive and would better illustrate the concept.

From a total of 3,473 compositions, 294 were studies so the remainder of forced mates (in 2, 3, 4 and 5 moves) was 3,179 or about 91.5%; still a sufficiently large sample, in my estimation. The number of mutual mates detected was only 29 or less than 1%. It is difficult to tell if this is low, average or high, but I suspected the frequency of mutual mates would likely be even lower in a sample taken from traditional (i.e., typically human composed) direct mate chess problems or a sample of forced mate endings from tournament games. A random sample of 3,179 direct mate problems by human composers (from a collection of 29,453 forced mates in 3, which I happened to have) revealed only two mutual mates or less than 0.1%.  Keep in mind that neither Chesthetica nor the human composers ‘intentionally’ composed these as mutual mates but instead with the stipulation that White is to play and force mate in n moves. It just so happens that in these 29 compositions, and two human compositions, if Black was to play instead, it would also be a mate in the same number of moves.

Also note that Chesthetica does not compose traditional chess problems but rather chess constructs (a type or class of chess problem) which means they do not necessarily have to conform to traditional chess composition conventions, e.g., not having a check in the first move. Put simply, chess constructs cover a larger spectrum of compositions and can lie anywhere between traditional chess problems and sequences typically found in real games. This casts the widest net in terms of aesthetics. Besides, if there’s anything I’ve learned about human aesthetic perception in chess over the years, at least primarily via the many chess communities on Facebook, is that people tend not to ‘like’ compositions they find too difficult to solve. 

In any case, the positions below show four examples of ‘mutual mates’ I chose from the 29 detected, and the two detected from human compositions. Incidentally, readers are also free to judge for themselves how aesthetically pleasing and thematic these are in contrast to the computer-generated ones. Try to solve for White first and then Black in the same number of moves, as per the modified stipulation. The solutions are provided at the end of the article.

White (or Black) to Play and Mate in 5
Computer-Generated Chess Problem 00726
Chesthetica v9.96 (Selangor, Malaysia)

White (or Black) to Play and Mate in 5
Computer-Generated Chess Problem 03224
Chesthetica v12.26 (Selangor, Malaysia)

White (or Black) to Play and Mate in 3
Computer-Generated Chess Problem 03302
Chesthetica v12.30 (Selangor, Malaysia)

White (or Black) to Play and Mate in 3
Computer-Generated Chess Problem 03312
Chesthetica v12.30 (Selangor, Malaysia)

Selection of Incidental ‘Mutual Mate’ compositions by Chesthetica

White (or Black) to Play and Mate in 3
Sam Loyd, Philadelphia 1858

White (or Black) to Play and Mate in 3,
CSL, The Chess Player’s Chronicle 1847

‘Mutual Mate’ compositions by human composers

The concept of a mutual mate or win should, I hope, be clear at this point. While the examples shown above may not conform to many traditional composition conventions or lack aesthetic appeal to some, this type of chess problem does not preclude any further conditions one may wish to apply. For instance, it could be specified to the composer that both the White to play and Black to play solutions should demonstrate the same theme and that neither should have any (major) duals either. It may also be specified that both the White and Black armies have the same amount or type of material. At some point, I would imagine a particular type of mutual mate may even rival the difficulty of creating a Babson task.

Mutual ‘wins’ (pertaining to studies), naturally, would generally be harder to compose compared to the direct mate variety, especially given that the wins should be decisive or clear at approximately, if not exactly, the same length. I have not programmed Chesthetica to explicitly aim for any of these, but if the idea catches on, I might at some point. Perhaps the reader, experienced composer or otherwise, would now like to try composing a mutual mate of their own.

Here are the solutions to the above examples:


We would like our readers to try their hand at composing Mutual Mate positions. Please submit any positions you might come up with to the editors.

Dr. Azlan Iqbal has a Ph.D. in artificial intelligence from the University of Malaya and is a senior lecturer at Universiti Tenaga Nasional, Malaysia, where he has worked since 2002. His research interests include computational aesthetics and computational creativity in games. He is a regular contributor at ChessBase News.
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Frits Fritschy Frits Fritschy 1/5/2022 10:54
I wish you all the luck. You haven't convinced me as I haven't convinced you. Let's leave it there.
azlan azlan 1/5/2022 12:02
@Frits: Master composers have many criteria they have set for themselves and that's fine. Other people who appreciate aesthetics in chess do not. Having said that, Chesthetica is still a better composer than 99.99% of the human population. Perhaps if I had the resources of a company like IBM or Google behind me, I could make it do even better. Alas, that may not happen so I wouldn't expect it any time soon if I were you.
Frits Fritschy Frits Fritschy 1/4/2022 02:56
Masters in any art may not have a monopoly on the term aesthetics, but it comes close to it. That's why they are masters in what they do and that's what they deserve for their hard work to become a master.
Any chessplayer, from whatever strength (leaving out absolute beginners) with a feeling for aesthetics will prefer the Loyd problem above any of the Chesthetics puzzles I've seen so far. If shown, or even more if solved by themselves. Quality always tells.
If you have programmed the concepts of 'long moves' and 'anticritical moves', Bh1 in problem two may well be unaccidental. Praise to you. Or, as I wrote, more or less (un)accidental, depending on whether you can establish a direct link between what you programmed and the choice the program made. If you did not program these concepts, it is accidental.
azlan azlan 1/4/2022 02:31
@Frits: Master chess problem composers do not have a monopoly on the term, "aesthetics". If you acknowledge that grandmaster players can appreciate aesthetics in sequences from their games (still far removed from master compositions), then you must acknowledge that even weaker players (e.g., club players) can appreciate it in theirs. In other words, aesthetics exists on many levels and is relative to the observer in most cases. Even chess problem judges often disagree with each other. Most chess players (including grandmasters) do not appreciate chess problems in the way composers do. It does not mean they have no proper conception of aesthetics (they certainly do in games of their level, for instance).

As for the Sam Loyd problem, I make it a point not to alter anything like this based on my own personal taste (I do agree that Nxc4 is the better variation, aesthetically, however). If two variations were given in the original database, I would have presented both and in the order they appeared. To be fair, I would also have included the next best line for Chesthetica's compositions. I'm glad even you found something of beauty in something composed by Chesthetica. If you cared to look through the many thousands of them already published in an unbiased way, I'm sure you would find even more. I can assure you, chance or randomness has little to do with it. Perhaps you can try throwing pieces on the board at random and see what comes up "accidentally" and how often it happens. You can even use the most powerful computers you can afford. I'll check back with you in a hundred years.
Frits Fritschy Frits Fritschy 1/4/2022 01:59
About the generated compositions in your last comment; I just looked at the first two. Not too bad, but they both lacked in economy (an idea shouldn't be expressed in 5 moves if it could be done in 4) and clearness of the idea. The key move in the first problem has a double threat and in the second problem you can use either knight for the check on e2. The two sacrifices in the first problem are rather crude, but I liked the second move in the second problem: there is some beauty there.

Okay, you produced software to create chess puzzles. Fine with me. Please don't connect it with esthetics before it is developed way further. Until then we might appreciate the rare moments of real beauty that more or less accidentally come up.
Frits Fritschy Frits Fritschy 1/4/2022 01:56
I wonder what you mean by 'the neurochemical experience of aesthetics'. It's not even clear whether there is a clear centre of beauty appreciation in the brain. Conflicting opinions can be found in two recent studies, and However, a clear warning against 'researchers attempting to reduce aesthetic experience to a set of physical or neurological laws': 'Overall, it can be argued that there is lack of proportion between the narrow approach to art taken by researchers versus the grand claims they make for their theories.' (quotes from

You gave a nice example of how esthetics ís enhanced by learning to appreciate complexities. For a beginner a smothered mate is something awesome, just because it ís way above their heads: they fall victim to it or get it explained by a stronger player, and after that they want to reproduce it. Same thing with the Saavedra position or the famous Réti manoeuvre in the pawn endgame. 'That's unbelievable', they exclaim, at the same time acknowledging they couldn't find it themselves and that they really appreciate it.
About the Loyd problem: (a) you should have checked it; (b) the line Nxc4 is the main line for anyone with any feeling for aesthetics; (c) this is not a Chesthetics composition; (d) if Chesthetics chose Kxc4 as the main line: see (b).
azlan azlan 1/4/2022 08:59
@Frits: I must disagree. Aesthetics (at least the neurochemical experience of it) is not necessarily enhanced by "learning to appreciate" complexities, even though it's possible. For instance, a beginner could just as easily have the same aesthetic "high" seeing a smothered mate for the first time. As for the Nxc4 issue with the Sam Loyd problem, the main lines were not chosen by me. I checked and it turns out that was the only line provided in the database I had. Still, it's ultimately up to the reader to analyze the variations for each problem. Only one variation is shown for all the other problems as well and not necessarily the most aesthetic one in any individual person's point of view, I'm sure. Chesthetica chooses its own main line for each of its compositions. Below is a selection of other compositions by Chesthetica which you might find more appealing. If not you, I'm fairly certain there are others who will. In fact, I know this for a fact.

Duplex problems were not invented by me, by the way. I'm just focusing on the direct mate variety in this article because they appear to be quite rare and difficult to compose. Especially where there is aesthetic appeal in the solutions for both sides.
malfa malfa 1/2/2022 10:57
@Frits Fritschy

You are both absolutely right. As far as I am concerned, the fairest thing I can say about this Chesthetica stuff is that its author badly needs a ton of advices about both chess and aesthetics.
Frits Fritschy Frits Fritschy 1/1/2022 01:49
Joshua, you can check the FIDE rating list for my gender...
JoshuaVGreen JoshuaVGreen 1/1/2022 12:52
@Frits Fritschy raises some good points, and with respect to the Loyd problem (s)he's correct that in the normal solution 1. ... Nxc4 should be taken as the main line.
AgainAgain AgainAgain 12/31/2021 07:14
Terrible as usual
Frits Fritschy Frits Fritschy 12/31/2021 02:34
"... people tend not to ‘like’ compositions they find too difficult to solve" I guess you mean that they don't hit the like-button in Facebook, which is something different from finding something pleasing. It is rather logical that you don't have an opinion about something when you can't experience it. And as most humans are just human, they will tend to appreciate something better when they have put some time in learning to appreciate it. And that's the whole idea about esthetics: great musical compositions may be harder for some (or many) to digest than supermarket muzak, but they are far more rewarding when you have learnt to listen to them. Esthetics is not about what the majority of the people do not dislike, it is about what at least some people really love. And it may even be something that most people would love.
Look at the problem by Sam Loyd. When you are really good at something, you can ignore some conventions, for instance a key move that gives check. In this case however, it's clear white can't do without a check, but there are many checks possible, and only one mates in 3. As the main variation, you give the least interesting one. I would nearly wonder whether you have seen what happens after 1... Nxc4. If you haven't (I hope and expect it was by accident), you shouldn't be allowed to write about esthetics.
That it is also possible to mate in 3 by black, is completely uninteresting, both from an esthetical and an educational point of view. Why should anyone, in the last case, care whether it is a mate in 3 or in 4? By putting Sam Loyd's name to it, in my opinion, you are spitting on his grave.