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Author: lievenlb

exotic chess positions (2)

Published March 5, 2008 by lievenlb

Juan de Mairena linked in the comments to last post to a truly great retro-chess problem ! In the position below white is to play and mate in three!



At first this seems wrong as there is an obvious mate in two : 1. Qe2-f1, Kh1xh2 2. Rg3-h3 The ingenious point being that black claims a draw after 1. Qe2-f1 invoking the 50 moves rule which states

The game is drawn, upon a correct claim by the player having the move, if
(a) he writes on his scoresheet, and declares to the arbiter his intention to make a move which shall result in the last 50 moves having been made by each player without the movement of any pawn and without the capture of any piece, or
(b) the last 50 consecutive moves have been made by each player without the movement of any pawn and without the capture of any piece.

I’d love to have the ‘official’ solution to this problem. Here’s what I’ve come up with, spending the better part of the afternoon… It will not be optimal but hopefully isn’t too far off. The crucial part is a maneuver unlocking the lower left-hand cluster of pieces (in particular the ordering of white and black rook). All captures were done with pawns and the final pawn move was b2-b3. Immediately before it the situation might look something like the situation on the left (essential is that the white king should not be too far from its home-square as he will be needed later to block the white rook)



after b2-b3 the bisshop on c1 travels to f4 and the black rook squeezes in to block the white rook so that also the black king can come in and position himself at e2. Then the two rooks evacuate the first row, allowing the black king to go to h1 and then the white king comes in to block the white rook from checking the black king (situation on the left below).



Finally, the white rook comes in and positions itself at e2, afterwards the white king evacuates the first row via b2 and travels to the right-hand upper corner entering via g7. Meanwhile, the black rook comes to g1, the white rook then travels to a3 and the black rook to a2. Then, the white king goes to b7 allowing the bishop to unlock the rook on a8 going to g7, allowing finally the king to go to d8…

Perhaps there is a much simpler and more elegant solution, so if you know, please comment. Oh, btw. how is the original problem solved. Well white first cancels the 50-move rule by 1. Kd8xd7 to continue for example with 2. Rg3-g4, 3. Qe1

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exotic chess positions (1)

Published March 4, 2008 by lievenlb

Ever tried a chess problem like : White to move, mate in two! Of course you have, and these are pretty easy to solve : you only have to work through the finite list of white first moves and decide whether or not black has a move left preventing mate on the next white move. This is even a (non-optimal) fool-proof algorithm to find the solution to this kind of chess problems. Right?

Wrong! There exist concrete positions, provable mate in two in which it is NOT possible to determine the winning first move for white! So, what’s wrong with the argument above? We did assume that, given the position, it is possible to determine all legal moves for the two players. So?

Well, some moves are legal only depending on the history of the game. For example, you are only allowed to do a castling if your king nor your rook made a prior move. Further, you can only make an en-passant-capture on the next move.

But surely all this is just theoretical? No-one ever constructed a provable 2-mate with impossible winning move. Wrong again. The logician Raymond Smullyan did precisely that in his retro-chess puzzle book Chess mysteries of Sherlock Holmes. Here’s the position :



Presumably every chess player goes for the mate : 1. Kf5-e6 2. g7-g8D But what if black counters your first move with a castling 1. … 0-0-0 ? Surely he isnt allowed to do this. Why not, is there any clue in the position to prove that either the king or the rook must have moved before?

Well, what was black previous move then? It cannot be the pawn move e6-e5 as before that move the white King would be in check, so what was it? Just one possibility left : it must have been e7-e5.

This offers then another winning strategy for white, as white can capture en-passant. 1. d5xe6 e.p. and then if black castles 1. … 0.0.0, 2. b6-b7 or is black does any other move : 2. g7-g8.

Hence, whatever the games’ history, white has a mate in two! However, looking ONLY at the given position, it is impossible for him to judge whether Kf5-e6 will do the trick!

Anyone seen similar constructions?

UPDATE

According to the wikipedia page on Retrograde chess analysis, the Smullyan-idea is an adaptation of a much older problem due to W. Langstaff in the Chess Amateur of 1922. Here’s the situation (the solution is the same as above)



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Finding Moonshine

Published March 2, 2008 by lievenlb

On friday, I did spot in my regular Antwerp-bookshop Finding Moonshine by Marcus du Sautoy and must have uttered a tiny curse because, at once, everyone near me was staring at me…

To make matters worse, I took the book from the shelf, quickly glanced through it and began shaking my head more and more, the more I convinced myself that it was a mere resampling of Symmetry and the Monster, The equation that couldn’t be solved, From Error-Correcting Codes through Sphere Packings to Simple Groups and the diary-columns du Sautoy wrote for a couple of UK-newspapers about his ‘life-as-a-mathematician’…

Still, I took the book home, made a pot of coffee and started reading the first chapter. And, sure enough, soon I had to read phrases like “The first team consisted of a ramshackle collection of mathematical mavericks. One of the most colourful was John Horton Conway, currently professor at the University of Princeton. His mathematical and personal charisma have given him almost cult status…” and “Conway, the Long John Silver of mathematics, decided that an account should be published of the lands that they had discovered on their voyage…” and so on, and so on, and so on.

The main problem I have with du Sautoy’s books is that their main topic is NOT mathematics, but rather the lives of mathematicians (colourlful described with childlike devotion) and the prestige of mathematical institutes (giving the impression that it is impossible to do mathematics of quality if one isn’t living in Princeton, Paris, Cambridge, Bonn or … Oxford). Less than a month ago, I reread his ‘Music of the Primes’ so all these phrases were still fresh in my memory, only on that occasion Alain Connes is playing Conway’s present role…

I was about to throw the book away, but first I wanted to read what other people thought about it. So, I found Timothy Gowers’ review, dated febraury 21st, in the Times Higher Education. The first paragraph below hints politely at the problems I had with Music of the Primes, but then, his conclusion was a surprise

The attitude of many professional mathematicians to the earlier book was ambivalent. Although they were pleased that du Sautoy was promoting mathematics, they were not always convinced by the way that he did it.

I myself expected to have a similar attitude to Finding Moonshine, but du Sautoy surprised me: he has pulled off that rare feat of writing in a way that can entertain and inform two different audiences – expert and non-expert – at the same time.

Okay, so maybe I should give ‘Finding Moonshine’ a further chance. After all, it is week-end and, I have nothing else to do than attending two family-parties… so I read the entire book in a couple of hours (not that difficult to do if you skip all paragraphs that have the look and feel of being copied from the books mentioned above) and, I admit, towards the end I mellowed a bit. Reading his diary notes I even felt empathy at times (if this is possible as du Sautoy makes a point of telling the world that most of us mathematicians are Aspergers). One example :

One of my graduate students has just left my office. He’s done some great work over the past three years and is starting to write up his doctorate, but he’s just confessed that he’s not sure that he wants to be a mathematician. I’m feeling quite sobered by this news. My graduate students are like my children. They are the future of the subject. Who’s going to read all the details of my papers if not my mathematical offspring? The subject feels so tribal that anyone who says they want out is almost a threat to everything the tribe stands for.
Anton has been working on a project very close to my current problem. There’s no denying that one can feel quite disillusioned by not finding a way into a problem. Last year one of my post-docs left for the City after attempting to scale this mountain with me. I’d already rescued him from being dragged off to the City once before. But after battling with our problem and seeing it become more and more complex, he felt that he wasn’t really cut out for it.

What is unsettling for me is that they both questioned the importance of what we are doing. They’ve asked that ‘What’s it all for?’ question, and think they’ve seen the Emperor without any clothes.
Anton has questioned whether the problems we are working on are really important. I’ve explained why I think these are fundamental questions about basic objects in nature, but I can see that he isn’t convinced. I feel I am having to defend my whole existence. I’ve arranged for him to join me at a conference in Israel later this month, and I hope that seeing the rest of the tribe enthused and excited about these problems will re-inspire him. It will also show him that people are interested in what he is dedicating his time to.

Du Sautoy is a softy! I’d throw such students out of the window…

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