In the picture a 3 by 4 chess board is presented. The goal of the white Queen is to force the black King to step onto the upper right corner square. The rules of chess should be obeyed.

This simple at the first sight problem after several unsuccessful trials may seem unsolvable to the reader. That's why we warn that the problem has a solution, but the solution contains a "subtle" move.

If you cought the essence of this solution then with no difficulties you can solve an analogous problem on an arbitrary rectangular board with sides of m by n squares. In all cases the corner square, where the King must be forced to, should be chosen prior to the game.

The problem is unsolvable only on square boards. Why?

M. MAMIKON ('OV' was added to the end of name by the editor)

[City Yerevan]

The first four moves 1. c3 a2 2. c1 b3 3. a1 c2 4. a2+ c3 pursue the goal to kick out the King to the right half of the board. (In case 4. ... c1 a trivial soluttion follows in threee steps.) Next 5. b1 d2 6. b2 d1.

In this situation the problem is analogous to the initial one with the only difference that the King must be forced into the nearest to him corner of the board. Here comes an unexpected "subtle" move of the Queen 7. a2(!) - stepping back from the King one square to the left. The following moves are obvious: 7. ... c1 8. b3 d2 9. b1 c3 10. a2 d3.

With the above mentioned 'fine' move the Queen stepping back from the King - begins the solution of the problem on any non-square board, when it's needed to force the King onto the nearest to him corner of the board. This problem is possible to solve for the King can not move to the corner diagonally opposite to the initial corner because of unequal sides of the board.

If the King must be forced onto the farthest corner, then he must be forced first into the diagonally opposite corner (the King himself runs to there). After this, the problem is to to force the King to the nearest to him corner which is being accomplished in the above described manner.

As for the problem on a square board, it is unsolvable for the following reason. The King moves to his diagonally opposite corner after which the situation is completely equivalent to the initial one due to equal sides of the board. As a result the King can always browse between the two diagonally opposite corners one of which is his initial position.

Occurrence of an interesting problem in a magazine usually causes a response - readers suggest answers, supplements, corrections. Many problems begin new life after the publication. They are being reprinted, included in collections, generalized, investigated. One of them appeared to be M. Mamikon's problem published many years ago. Recently we have received a letter from its author...

In 1976 I called in at your editorial offices and handed over three problems which I had created. They were published in # 4 for 1976 in the 'Psychological Practicum' section. One of the problems was as follows: the white Queen is to force the black King to step into the square marked with a question in the upper right corner of this mini-board (d3).

It was also warned, that the problem had a solution and if the reader caught the essence, he would be able to solve a similar problem on any rectangular board with the sides of m x n squares. At the end a question was asked. Why there is no solution in the case of a square board?

The solution of this problem contained a quite subtle move. Everyone can drive the king to the bottom right corner (d1)

1. Qc3+ Ka2 2. Qc1 Kb3 3. Qa1 Kc2. 4. Qa2+ Kc3 5. Qb1 Kd2 6. Qb2+ Kd1.

The "subtle " move of the Queen is the retreat to a2. Now it is obvious that the king is forced to step onto the required square with a question mark (d3) in four moves:

7. Qa2(!) Kc1 8. Qb3 Kd2 9. Qb1 Kc3 10. Qa2 Kd3.

The detailed solution of the problem was given in # 5 for 1976 in the "Answers and Solutions". There was also given the proof that the problem is solvable on any rectangular board and can not be solved on a square board.

Within several weeks after the problem was published, the readers sent letters, in which another graceful solution was suggested: the self-sacrifice of the Queen. In the situation just preceeding that on the second picture the positions of the Queen and the King were b1 and d2 respectively. Here the Queen moves on the marked square 6. Qd3+ offering a sacrifice. If the King accepts the sacrifice he steps onto the required square and hence loses the game. Though the Queen does not exist any more, the whites win according to the condition of the problem! Certainly, the King does not accept the sacrifice and moves to c1. But in this situation the subsequent moves are obvious and lead to the Queen's victory. The total number of moves in this case is one less than in the previous solution which was equal 10. The sacrifice of the Queen can be also offered on other chessboards.

Since "Science and Life" first published this problem, over 25 years passed. It underwent (has lived) an amazing history. First of all I must say that I was very proud of it and played this game, probably, with thousands of people, enjoying their constant failure. I even played it with Gary Kasparov. He came to Armenia with his mother to visit his aunt, who worked at the Institute of Cardiology. My friend, cardiologist Ashot Davtian, drove them to Byurakan Observatory, where I had worked from 1965 to 1990. Then Gary was only seventeen, and he just became the champion of the USSR among the youths.

We took a walk in a beautiful park of Byurakan Observatory, and then sat down on a stone bench near a small fountain. I didn't miss the opportunity to play with him my mini-chess game. I drew a 3x4 chess-board on a piece of paper and put two coins on it for the King and the Queen. Gary was confused with such an unexpected formulation of the problem, and, after several moves he said, that didn't see any solution. "Is it possible to solve the problem?" he asked without any serious intention to finding a solution. I immediately suggested that we should exchange the pieces, and I won the game. I was very happy with such a turning point because I won a chess game against Gary Kasparov himself! And then I told him: "You lost, but I want you to become the world champion ". And then asked: "Can you do that? Do you promise to become world champion?" Gary overcame his modesty with difficulty and said in a low voice: "Yes". And he became! And there is also my service in it. I hope, Garry will recall this story and will not take offense at my revelations.

Now I live in the USA and work at Californian Institute of Technology. At leisure I still make up different riddles and puzzles, but usually do not publish them. But I publish articles offering elementary solutions to problems of integral calculus, or so called "highest mathematics".

I have even created a computer game where it is possible to play my mini-chess game with the computer. See it on the Internet at

http: // www.its.caltech.edu/ ~ mamikon under the heading "Games", named "Mini-Chess".

On this website it is possible to find many other games and simplified solutions to problems of highest mathematics which I created.

But the most interesting story is ahead. (I feel as if I have completely forgotten Russian during the fifteen years, spent in USA. Haven't I? Thanks!)

In 1987 the problem was published in the American magazine "Games", or rather the last part of it (beginning from the second picture presented above). Exactly same notations at the squares, including the same upper right corner, where the King should be forced to. Moreover, the published solution repeats those four moves which I described in the "Science and Life" # 5 for 1976. It is necessary to mention, that the publication was offered by Marek Pensko &endash; a well-known Polish puzzle-maker and writer, the president of the Polish branch of the "World Puzzle Federation". In the foreword to this publication he claims, that the problems (there were also two other problems) had never been published before.

Famous Martin Gardner who can be called today's "king of puzzles" and the professor of mathematics of the University of Alberta in Canada became interested in this problem and generalized it to the case of any m x n board. They published an article in the "Quantum" magazine which is but the American version of Russian well-known magazine "Quantum".

Many employees of "Quantum" moved to USA and began to publish an American version on the basis of the Russian magazine, in cooperation with the Russian publication. Compared to other American magazines of the same style, it was rather good and popular. It has been closed for financial reasons about two years ago.

And so, they published an article titled "A Royal Problem", considering the idea of the solution on any m x n board. In a month three students engaged in the Math Club, lead by the Canadian co-author Andy Liu, attacked the general problem and published an article called "Martin Gardner's Royal Problem" in the same magazine. All this was published in the issues for July and September, 1993. None of them gives a reference to previous authors, including Marek Pensko. Certainly, I didn't know about this at that time.

After 8 years, in 2001, a Martin Gardner's book "Gardner's Workout" was published, where exactly the first article from "Quantum" with several additions was presented. Gardner mentions that he generalized a particular puzzle of Marek, published in the "Games" adding "the date of which I have lost". I neither saw this book nor knew about it.

A year later my friend suggested that we should go to Atlanta (Georgia) where once in two years Martin Gardner's friends arrange a conference called "G4G#" ("Gathering For Gardner #"). The number indicates the current number of the conference. I went to the conference "G4G5" in 2002 and carried "flyers", which I had prepared: two pages of my problems and puzzles. Each participant brings his riddles, all of them leave conference with a "bag" of puzzles. Among my puzzles I suggested that very mini-chess game, also pointing to the date of its first publication (though I made a mistake saying it was published in 1975 instead of 1976). Nobody even paid attention to the fact that I had "stolen" this Martin Gardner's problem, published in magazines and in his book.

I found out this story by chance two weeks ago. And it is good and bad. It is good, that my problem became "celebrity"; it is bad, because it's not on my behalf. I'll certainly write about it to the "Games" ("Quantum" doesn't exist any more), and also to Martin Gardner and his colleagues to Canada.

The most interesting thing is that the surprising subtle solution consisting of the sacrifice of the Queen, which the Russian readers found in 1976 in a week, remained unnoticed.

I want to take advantage of these correspondence and tell the Russian readers that in the Soviet Union they were the most educated and the most inventive people in the world. I want them to value these qualities and never lose them under any circumstances.

Future belongs to a country of wise people... Technology tomorrow will become outdated, but knowledge, creativeness and intellectual skills will live forever.

Sincerely yours

M. Mamikon,

e-mail: mamikon@caltech.edu

Pasadena, California, August 2003

___________________________________

Adding to M. Mamikon's interesting story, we should also mention the fact, that his problem was also included in Ye. Gick's book "Chess and Mathematics" (1983). There is also no reference to the "Science and Life", though in the end of the book the list of the used literature is presented. Many people understand this it as a list of recommended additional literature for reading, not as the used ones.

Republishing is not always a plagiarism. The problems which are published in the popular sections usually have two authors: the author of a problem and the author of the text. The author of the text presents a problem, without pretending to the priority - his goal is to popularize it. The readers of our magazine are familiar with such remarkable popularizers as Y. Perelman, B. Kordemsky, M. Gardner, etc...

Professor Ya. A. Smorodinsky - the member of the editorial board of "Science and Life", a big fan and an expert in the entertaining popular scientific literature, said, that M. Gardner does not, normally, create own problems. As a matter of fact it was a great honor to an author of a problem (whether he was an amateur or a well-known mathematician) to be quoted in his articles in the section of "Mathematical Games". Many puzzles gained worldwide popularity due to Gardner's interpretations.

During a long period Marek Pensko conducted the section of "Logical Games" in a popular Polish scientific magazine "Problems". He replaced Leh Pianovsky in 1974, an enthusiast in popularization of logic games and puzzles, unfortunately left untimely. The "Problems" acquainted the readers with the best samples of world popularization in the field of mathematical knowledge and logic.

B. A. Kordemsky in the foreword to his book "Mathematical astuteness" (1965, eighth edition) informed the reader of the following: "The significant part of the problems and their subjects is obtained by me from the books and magazines..." Further a list of names was given ending with "etc...", used frequently in such cases (i.e. the names not listed).

After all, the result is important. M. Mamikon's problem was noticed. It is interesting, and it was the first publication. The author can be pleased. But perhaps somebody else came across this problem earlier. History will correct if soŠ

I. KONSTANTINOV

The Full Story of the Mini-Chess Game (by Mamikon)

Introduction

In 1964 when I was a graduate student at Yerevan State University in former Soviet Armenia, there were yearly chess championships within the university. My classmates had a great believe that I must be the best chess player because I was "extraordinary" in mathematical and physical subjects. But I disagree with such a common point of view. For me playing chess was a nightmare, and I had not much experience. The reason is that I can not make a move unless I can see its full consequences which is impossible in chess because nobody (even computers today) can analyze chess till the end. Therefore every move seems stupid to me. The result of my friends' efforts to engage me into the championship turned to be very sad and created a controversy between us.

To prove to them that nobody really knows how to play chess and that each their move is 'stupid' I decided to create an example that will fully demonstrate my point. I know that the most confusing situations are the unusual ones where nobody has any experience or habits. To create such an example I had to start with the simplest possible situation - the smallest board and minimal number of figures. And I had an idea which is the only idea in chess that I knew and liked. It is the well known 'tempo' tactics, when one repeats the exact same situation on the board but with the player's turn is exchanged .

First I tried just 3 x 3 board with a King and Queen to see if I can do something with stalemate situation. It did not work. In a week or so I looked back at my ideas and expanded the board so the figures can have some moving options. I didn't know what I wanted but I had a clear goal to demonstrate that the chess players can not see the well known 'tempo' trick in unusual though much simpler situations. Suddenly it became clear to me that I should target the corner! What a surprise!

So, the simplest thing to do was to stalemate the King, but the King has a choice of the corner on the shortest side of the board. It was tricky to push the King to a prescribed corner along the shortest edge, and here was the tempo trick necessary. Although for the player such a move seems as the stupidest of all that was the only move leading to the solution. I was lucky to be able to combine several ideas into one paradoxical complex working perfectly in its quintessence. Naturally I tried other boards. Because 3 x 3 did not work for my game I was curious to try the 4 by 4 board. It wasn't working. The King has even more choices of corners on any square board I tried. But the goal of the game was reachable on any other non-rectangular board with the 'tempo' trick. It was even more interesting and confusing for my 'enemy-friends'. I played this game with hundreds of friends and other students. It was so enjoyable that became my preferred topic in any meeting with people regardless of did I know them or not.

Most surprisingly, during many such games I noticed another possible trick which was achieving the goal even faster - shorter by one step. This is my main point, which seems missing everyone's attention. I want to make it clear after 40 years of a very interesting history of this game.

There was another philosophical point in my game.

Those years Michael Botvinnik was world chess champion He was a computer science engineer and has designed first chess programs for computers. He held the champion's title for over 15 years as a hobby. It was then that the Soviet computer (I think it was called Kaissa, the goddess of chess) won against the American computer. It was the very first series of chess games between computers in history. The soviet computer bit the American one, but the most interesting fact was that in one of this games (the fourth in that series) the American computer did not see a one-step checkmate and got it to everyone's surprise. It gave up only after it was checkmated! How primitive it was?

From a former student of Botvinnik (Peter, I forgot his last name) whom I worked with in Byurakan Observatory, I learned some ideas on the algorithms that Botvinnik has implemented. It is very important to realize for people who think that computer will have an advantage before the mankind because it's faster. I don't think this is necessarily true for the following reason. The computer calculates all possible situations, say, after 5 moves. But then it has to make a choice - which situation is the most preferable. And this has to be based on quantitative scoring of the situations. There should be a numerical value for each situation on the board so that the computer will take the highest ranking one. This analysis of figures on the board can be done only intuitively due to experience of talented chess players, and this evaluation can not be uniquely defined and more could be wrong as well. And that my point of view I was supporting with my mini-chess game. Imagine that after 7 steps computer came up with all the possible situations on the board one of which corresponds to the situation just after the stupidest move in 3 by 4 board, or this can be a part of a bigger-chess picture. What is computer going to do? It will miss the only correct move because no program-designer knows it. So it will lose. All other normal moves which will be implemented as smart ones by the experienced players in fact will be all stupid. The only smart move is the "stupidest' one. And that proves my point. Actually, I doubt that computer can win against me in my game, because, in this unusual game, nobody has my experience to put his intuition into the algorithm. unless of course he learns from me. So it is quiet possible to have an extraordinary and unusual player with some crazy intuition and strategic thinking who will win against the supercomputer. All this is true because the chess game can not be yet analyzed to its end! I doubt if it can be ever done. When it happens then no intuitive evaluation is needed in the program. The computer will choose without any analyses the shortest play which leads to a checkmate. It's that clear... What I am really saying is that playing with computer is a game against all previous chess masters' experience, but all intuitions can be bitten by another super-ingenious chess-talent with a deeper and crazier intuition (Bobby-Fisher-like). I am for a man unless chess is analyzed till the end.

Strange facts making "Quite a Detective Story, A Puzzle". Can you resolve them?

In this entire story, I highlight to emphasize a number of facts which seems so unususl, curious, coincidental, strange and some even 'suspicious'.

1. Why would anyone come up with such a 'meaningless' chess game? This can be answered only by the real creator.

2. Marek is from Poland which is next to Soviet Union, and Russian language is so close to Polish language to understand, also it is commonly used.

3. Ye. Geek used my problem in his book "Chess & Mathematics" (1983) - another close source easily accessible to Marek.

4. Labeling and the upper right corner, as well as Marek's solution, literarily copy my labeling and solution.

5. Why did Marek claim and (more strangely) why did 'Games' magazine publish Marek's claim that these problems "have never been published before..."?

6. Incidentally, Gardner takes Marek's puzzle back to my original both particular and general settings.

7. Gardner publishes the article in a pro-Russian magazine "Quantum" - leaning back towards the original Russian source.

8. In his 2001 book "Gardner's Workout" Martin Gardner refers to Marek's publication in 'Games' magazine saying "the date of which I have lost". He is on the board of that magazine, so by a single call he could find out the date.

9. In Atlanta in 2002, at G4G5, my exchange gift was a flyer which included my mini-chess problem. Anyone could mention that it was the "Martin Gardner's Royal Problem", but nobody brought this up.

10. All American publications missed the shortest, therefore the only, correct solution - with Queen's sacrifice. This solution was published in Russian not in the second issue but later, in the third issue, so it would be very naturally overlooked.

How To EXPLAIN these facts?

The only way to check YOUR answer to these question is to ask Martin Gardner, the Sphinx.

Here is MY logical explanation:.

Martin Gardner has always been extremely careful to refer to the source of puzzles and problems he writes about. He also has an extensive knowledge of existing and out of print publications on the history of puzzles. He certainly knew about the "Nauka i Jizn" Psychological Practicum" section and most likely saw the Mini-chess problem with its solutions in 1976. It was maybe interesting to him, but so what? Unless you analyze it in detail you won't get involved into it.

Gardner was on the board of contributing writers of Games magazine where 11 years later a puzzle "Cornering The King" was submitted by Marek. His opinion was vital in publication of such issues. Something rung in his head regarding this idea, something was so familiar to him. Did he see it before? He was almost sure that it was published somewhere but he could not remember where (after 11 years!). So, he suggested to ask the author if the material is original. You know how sensitive are the publishers about copyright issues. Marek replied to this request in writing with his claim "these brainteasers were made for GAMES and have never been published before". The editorial board was not truly satisfied with this claim and took a precaution by publishing the claim 'on behalf' of the author.

The submission of this puzzle, in which Gardner had to be involved and engaged in its analysis as a referee on behalf of Games magazine, was a refreshment that subconsciously triggered a question: what about a general case of arbitrary board? Now there was a reason to think about it deeply because the general case was not considered and seemed interesting. So, after a brief analysis he posed a question in The Journal of Recreational Mathematics, and started investigating the general problem thoroughly. The results were the two articles published in Quantum.

Gardner intuitively felt the problem to be of possible Russian source, and chose Quantum for that reason. Could a Russian reader shed some light? It was not to be. (Ask me "why?"). I thinkGardner, nonetheless, had doubts that Marek was the original creator; something was bothering him. Maybe Gardner's intuition caused him to ignore references to Marek in Quantum. In the 2001 book Gardner's Workout this issue rises again as he presents the problem again: whom to refer to? A book is a more serious and more complete matter than articles, it's more summarizing, permanently fixed, and references are more than desired. Gardner needed a reference because he was using somebody else's idea, he had to be honest as he has always been. Being forced to refer to the only source known to him, he mentions Marek - just to close the case. But because Gardner's sense of fairness has always been strong, his intuition defies him and he writes: "the date of which I have lost".

M. Mamikon

P.S. FOR CONFUSED MINDS:

I DON'T BLAME MARTIN GARDNER, TO THE CONTRARY - I PRAISE HIS WISDOM.