1/8/2016 – QM3: A game ends with the move 6...Nf1 mate – find moves that achieve this. That was our New Year's Puzzle. Solutions were sent in from around the globe – hundreds in all. Many letters were quite interesting, so we bring you a substantial selection. One of them has won the historic prize. Finally the coin problem by John Nunn: the solution is a five-second eight-word sentence that any ten-year-old can understand.

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On the first day of the year we published a prize puzzle for our readers to solve. There was a chance to win a prize that has some historic significance: the chess engine Hiarcs 13 with a signature of the 13th World Champion Garry Kasparov.

The problem by was the following proof games/quick mate problem submitted by Stuart Rachels:

QM3: A game ends with the move 6...Nf1 mate |

It is a black move that ends the game, and, very importantly, *it is not a capture!* Our test candidates came up with moves that end in 6...Nxf1 mate, which is fairly easy (but wrong).

When we published the problem last Friday we mentioned that one of the world's leading problem solvers and experts in exactly this kind of puzzle was still struggling to find the solution. The expert, as many had guessed, was John Nunn. Shame on him?

In the meantime we found out why John was not showing his usual instant solving ability: he had assumed that the moves leading to 6...Nf1 mate had to be unique, as they were in a previous problem ("A game ends with the move 7.Ka3 mate"). So he immediately ruled out any lines where variations were clearly possible – e.g. moving a bishop to c4 or b5, or playing the same moves in a different move order. When we told him that was allowed the solution came within minutes.

We asked you to send in solutions using the email feedback at the bottom of the page, and our inbox was soon filling up. Many of the messages were quite interesting to read, and so we will share a selection of them with you. In each case we will leave out the moves of the actual solution so that you can still try your luck with the problem. After reading some of the letters this should no longer be impossible.

**Zhichao Li, Zhuhai, China** (the very first message to arrive)

It took me about 90 (ninety) minutes to find the solution. First it is easy to show the knight is not a promoted pawn, since it takes five moves for a pawn to reaches the first rankand another two to move to f1-square. First I tried to find a solution where the white king ends on the g3 square, unfortunately the last move is a capture: 1.e4 e5 2.Ke2 d5 3.Kf3 Nf6 4.Kg3 Ng4 5.Qf3 Ne3 6.a3 Nxf1 checkmate. I tried to improve this solution but failed. Then I thought about double check in the last move, with queen on d3, h3, g5, c7. Unfortunately they all take seven moves. When I tried to find the double check with queens on g5 and c7, I come to the idea that White can castle. For example: 1.g4 Nf6 2.Bg2 Nh5 3.Nf3 c5 4.0-0 Qc7 5.h3 Ng3 6.Kh2 a6 7.Rh1 a5 8.Ng1 Nf1 double check and mate. Unfortunately it takes 8 moves. Then suddenly I have the idea... . Thanks for this wonderful problem, and a Happy New Year!

**Eric Black, Illinois, USA**

I spent 32 minutes on the problem. I tried to mate the white king on d2 initially. I realized there were not enough moves. I then tried to mate the White king on h2. Again not enough moves. I then thought the solution must be a discovered checkmate. I had four black knight moves but only two other black moves to utilize. *[Correct thought process then given].* I found the solution right away after that.

**Laszlo Antal, Füzesabony, Hungary**

It is always a pleasure to solve the Christmas and New Year’s Puzzles. As requested, I measured the time I needed to solve the puzzle. I read the ChessBase article at 8.20 a.m. and started to think about it without a board. Around 8.40 a.m. I set the pieces on a chessboard and tried to solve the puzzle. I admit that sometimes I moved the pieces. By 9.20 a.m. I had found the solution, so it took exactly an hour.

**Tobias Nordqvist, Sandviken, Sweden**

You might not believe this but I solved the problem in under one minute! Yes, it's almost like I can not believe it myself. [Gives a solution with 6...Kxf1#, and then follows up with:] Sorry, I must have gotten hypnotised. I must now go out to watch bandy match, but I will get back to you if .... I find the soulution. Probably I will have to do as Nisha Mohota and sleep on it . Hope the bandy match now not will be completely ruined by thinking about this solution instead of enjoying the game. – *[The next day, together with the correct solution]:* In Sweden a working day is eight hours, and it was almost as long as it took me to solve this problem. Please ChessBase stop terrorizing us with these problems during our vacation. Now my free day is gone. It would be much better if you put these problems during the regular working time.

Bandy match? No, we didn't know either, but were intriqued to find out. Bandy is a team winter sport played on ice, and is the second most popular winter sport in the world. It is played on ice like ice hockey (the most popular), but like field hockey, players use bowed sticks and a small ball. The bandy field is about the same size as a soccer pitch (!) and like football the game is normally played in halves of 45 minutes each, with eleven players on each team. Here's some video for you to get an impression:

Great stuff, don't you think? Maybe better than ice hockey? Thanks Tobias for showing it to us.

**Niels Lauritsen, Copenhagen Denmark**

A bit tricky, since I started with two assumptions, both being wrong. At first I thought I should look for a discovered double check and mate, such as with white king on d2 against the battery of black knight on e3 and queen on g5, giving discovered double check and mate by Nf1. However, it takes seven moves to plug all the escape squares for the white king. So after 20 minutes I turned to alternatives. *[Correct solution given].*

**Mayur Gondhalekar, Tokyo, Japan**

I finally managed to solve the New Year's puzzle – it took me one hour. Initially I ended up solving for 6...Ng1 mate, and almost sent in the solution. But I realized my error and retried. This took up 20 minutes.

**Nisha Mohota, India**

I must have taken around five hours totally (night and morning combined) on this problem. I must say that I must have been crazy to take so much time. Now that I have solved it I am asking myself "REALLY? I got stumped by THIS problem?" I still can't believe it: at night I took one hour thirty minutes (but I can maybe excuse myself as I was drowsy). But two hours fifteen minutes + the time I kept thinking after going to terrace or lying down or eating today morning sounds ridiculous! I was thinking of all crazy ideas to mate and not this really simple one! This is what happens in actual tournament games, even by very strong players: they go deep in their calculation and overlook one-move tactics in between. My brain really needs to be polished, it has become dull; missing simple ideas! What do I do to make it sharper? The only thing I dislike about this problem is that it has multiple moves – not only moves like in other such problems!

**Matthew Anderton, United Kingdom**

The solution to the New Year's puzzle is *[correct solution given]*. There are other move orders, which is slightly disappointing as other puzzle of this type require a precise move order. But it is nice that White has just developed 'normally' – almost!

**Zoltan Gyimesi, Budapest, Hungary**

I might have missunderstood something, but it looks to me that the 6....Nf1 mate puzzle has multiplie soultions.

**Uberto Delprato, Rome, Italy**

A as always I do enjoy your Season's Puzzles, they are great fun during the few days of the year when I have some time to spend at the chessboard. I took up the QM3 challenge today and to my surprise I found a solution in less than five minutes. I spent almost an hour looking for other solutions and, more importantly, if my solution was actually working. Well, I could neither find alternate constructions nor convince myself that I was wrong.

**Jimmy Mårdell, Sweden**

Very nice problem! Took me 2-3 hours. I had been looking at other kinds of discovered checks first along various diagonals and ruled them all out. Then I found so many nice mating patterns with the white king on e3, d2 or g3 that were just one half move short. That got me so frustrated.

**John Tromp, Long Island**

Having looked at the problem throughout the day (total about one hour), I just found the delightful [solution given]. I soon recognized that a discovered check is pretty much necessary as it takes too many moves to make Nf1 uncapturable. After exhausting the diagonals, I started looking at rows, but these all fell a move short. The last resort was along a file, and all pieces fell magically into place!

**Lasitha Herath, Sri Lanka**

Tough!! was really tough!! It took me about six hours! Tried all options where Nf1 is a check by knight, with the king on h2 and d2. Then tried a few with Kg3 and Ke3 with opened h files and d files. Even tried promoting a pawn on f1 for a knight. Nothing worked. Discovered checks with Qb6, Bh6 were useless. Then only I realised *[correct solution given]*. Gotcha!

**Gaston Franco, Buenos Aires, Argentina**

First of all I would like to wish you a Happy 2016, and to thank you for the puzzles (and prizes) that I look forward to every year for quite some time now. Even though I had solved quite easily the first two Quick Mate problems, I found the quick mate #3 really hard, and spent a long time looking for tries like Qe1xe7 ...Qd8xe7 or Bf1-b5-xd7 ...Qxd7 (or ...Bxd7) freeing the black queen without having to move a pawn first, none of which seemed to work. Suddenly, after thinking of a lot of other discovered check ideas (with a black queen on g5, for instance) I happened to think of the possibility of castling, after which it was easy to find the solution. I felt a bit disappointed having taken that long to find it. Fortunately, I solved the coin problem almost instantly...

**Bryce Tiglon, Redmond, WA USA**

It took be about 10-15 minutes and I solved it by putting the N on f1 in four moves and then trying out multiple other two-move combos. Then suddenly it hit me! The king didn't have to be on d2! I got the answer almost as soon as I realized this.

**Sergei Kaunin, St. Petersburg, Russia**

It's taken me three hours yesterday and half an hour today morning. I found the solution by means of analyzing all possible mates with the knight on f1. When I decided that first visible mates taken more than six moves I probed the 0-0 variant, which was succesfull.

**Dallas Gatti, Melbourne, Australia**

This puzzle took me about 15 minutes. Initially it was about trying to lock the White King on d2 and deliver mate, but quickly realized that blocking the bishop, rook and queen from the f1 square was impossible. Moving the king to e3, g3 or h2 and lock it in as well as blocking all squares for 6...Nf1# was impossible. So the solution was a discover mate. Quickly realised king in the centre was not the solution, which only left that white had to castle. Once I got onto this theme I solved it within a minute. I really enjoy these puzzles, which I was introduced to by the 1999 Christmas Puzzle. I didn't solve it, but got to study proof games, where a certain position is achieved in a number of moves. This is not a proof game, as there are many ways for this puzzle to be achieved (the pawns can move either 1 or 2 squares, the bishop can move to any square other than e2, etc. But was an very enjoyable puzzle.

**Ludo Tolhuizen, Waalre, The Netherlands**

*[Correct solution given]* Of course, many variations are possible (move orders, pawn to e3, bishop to d3, b5, etc) Costed me 2-3 hours. Logical puzzle cost me about five minutes.

**Clive Frostick, Surrey, England**

It took me most of the day to solve this problem, thinking about it on and off. Perhaps a couple of hours working at a board. Mostly I wasted time trying to get other mating positions to work. The hardest part is spotting the castling option, which seems unlikely because it puts a rook back on f1, the very square we are working so hard to clear. Certainly it is harder than QM2, and the fairly easy QM1. Thanks for another Christmas treat over the past few days!

**Jack Findley, The Woodlands, TX. USA**

Thought about it for 30 minutes yesterday and then another 30 mins today when I finally got it. At first, I thought it had to be a discovered double check so I set up various mating positions with the King at h2, d2 and even g3. I was able to get those with seven moves for white, but not 6. Then I realized that if I could remove the h pawn and put the King in the corner, rook on g1, a simple discovered check with the Black Queen would do it. When I was able to achieve that setup in only 6 moves, I knew I had solved it. The coin problem only took 2 mins.

**Manikumar, Chennai, India**

Was trying initially for a discovered double check with bishop or queen. Finally only got the idea of castling. But overall took only a few minutes. Is it really true that senior persons are finding it hard? Thanks to one of my WhatsApp group, which primed me towards helpmates. *[Correct solution given]* That's one of the few solutions. The sequence of white moves must be the same. But the first move can be e4/e3 and the second move can also vary. For the Black moves, only the last two moves must be the mentioned ones. The sequence of the first four moves can be interchanged.

**Karlo Gavranovic, Borås, Sweden**

Time: 1 h 21 min. I "knew" it was about discovered checkmate but I spent almost all my time on the variants with black queen on d6, knight on g3 and white king on h2 but could not find anything in six moves. Finally, I got an idea of castling and I first tried to exchange black h-pawn and use R + N before I saw that it works better with queen. By the way, I spent a whole afternoon on QM2 (7.Ka3#) without finding a solution).

**Shaunaq Marathe, India**

I was coming back home after a gruelling MBA entrance exam called XAT on 3.1.15 in a Mumbai local when I looked at this article. It took me close to 50-60 minutes of thinking without a board in the crowded train to solve this. *[Correct solution given]*. I hope I am correct. Would love to hear back from you even if I am wrong or not the first one to send the solution.

**Koosha Jaferian, Iran**

I got stuck for a long time on trying to mate the white king on d2 with double check by knight and bishop! But then found the right solution! *[Bonus problem also solved]*. Thanks a lot for you great problems!

**Koen Eeman, Denderleeuw, Belgium**

I nearly immediately found puzzle five as well as the coins puzzle *[correct solution given].* but boy did I suffer on puzzle six!! I discarded the option to mate the king in the corner quite soon (miscalculated) and found myself struggling over the weekend to make it work, mainly with the king on e3, which I can now safely confirm is not possibly in six moves. I add some of my relentless fails herewith for your amusement.

[Event "Failed attempts"] [Site "?"] [Date "2016.01.03"] [Round "?"] [White "Koen, Eeman"] [Black "6...Nf1#"] [Result "*"] [ECO "B04"] [PlyCount "12"] [SourceDate "2016.01.06"] 1. e4 (1. d4 Nc6 2. e4 Nxd4 3. Bb5 Nf3+ 4. Ke2 Nxh2 5. Bxd7+ Qxd7 6. Bf4 Qxd1+ 7. Ke3 Nf1# {Nice one with Nf1 mate on move seven!}) 1... Nf6 (1... e5 2. d4 Nf6 3. Nc3 Ng4 4. Bd3 Ne3 5. Kd2 Qg5 6. Nge2 Nf1+ {Oops e1.}) 2. Qg4 (2. Bb5 Ng4 3. Bc6 (3. Ke2 Nxh2 4. Bxd7+ Qxd7 5. Ke3 Qd6 6. Qf3 Nf1+ {Oops e2.}) 3... Nxh2 4. Qg4 dxc6 5. Ke2 Bxg4+ 6. Ke3 Nf1+ {Oops f4.}) (2. Bd3 Nh5 (2... Ng4 3. Ke2 e5 4. Kf3 d5 5. Kg3 Nxh2 6. Qf3 Nf1+ {[%cal Gd3f1] Bad bad bishop.}) 3. Ke2 Ng3+ 4. Ke3 e5 5. Qf3 Qh4 6. Ne2 Nf1+ {Bad rook.}) (2. Ba6 Ng4 3. Ke2 Nxh2 4. Bxb7 e5 5. Ba6 Bxa6+ 6. Ke3 Nf1+ {Oops f3.}) 2... Nxe4 3. Qxd7+ Qxd7 4. Ke2 Ng3+ 5. Ke3 Qg4 6. Bd3 Nf1+ {Bad bishop.} *

**Juha P., Helsinki**

Thank you again for the entertaining holiday puzzles! The 6. ... Nf1 mate turned out to be quite straightforward after I realized a couple of key points. First it feels unlikely that Nf1 is delivering the mate, since it is difficult to "remove" the white defenders (queen, rook, bishop) of the f1 square. This lead me to look at discovered checks, where it is quite obvious that a black queen is needed to deliver the check. In this case the white king cannot be around e1, since it would be able to simply capture the knight. What remains is to look at discovered checks where white king is in h1 and last black move is 6. *[Correct solution given]. *What bothers me with this solution, though, is the fact that there are multiple alternatives to the first three moves, so 1.e3 e6 and 2.B to any square beyond d3 should work...

**Michael McDowell, Westclliff-on-sea, England**

*[Correct solution given]* Other move orders are possible, and there are many alternatives, e.g. the bishop can move to other squares, and the knight can come out via h6 instead of f6.

**Ben Hague, New Zealand**

Took me about three hours in total, just didn't consider this checking setup at all until around 2:55 hours in. Did come across this game in the process: 1.e4 Nf6 2.Bd3 Nxe4 3.Qg4 Nxf2 4.Qxd7+ Qxd7 5.Ke2 Qg4+ 6.Ke3 Nd1#. I'd imagine that is the quickest ending in 6...Nd1#.

**Mark van der Hoorn, Wellington**

I like these sort of puzzles, but this one was tough. How I solved it was probably the normal way. I tried to figure out the fastest way for black to play Nf1 and then worked out how many spare moves I had. The next thing is to pick the checkmating square. If the knight is giving mate then there are four. And it takes four moves for a knight to reach f1. So that leaves two moves to cover the escape squares. The real problem is that f1 is difficult to leave unguarded. This suggests that a discovered mate is the way to go. I tried a few, but everything seemed to take one move too long. Eventually I hit upon the idea of the white king being on h1 and then it was simple. Too simple really. The white e-pawn moves, the bishop goes to d3 or beyond, he white knight to e2, and then White castles, puts the king on h1 and the rook on g1. Black moves the e pawn, puts the queen on h4, the knight takes on h2 and then Nf1#. The thing is, there seems so many different ways to do it, but only the one idea. I'm not complaining though. However, while trying to find the solution I came up with a few interesting ideas. My favourite one goes like this.

**Peter Finn, High Wycombe, England**

Finding this solution took me approximately three hours. At first I looked at the possible final positions of whites king and pieces. However it quickly became evindent that if the knight check wasn't a discovered check then either white takes too many moves to entrap his king or black ends up taking the bishop on f1. Then I started looking at discovered checks. At first I looked at the possible positions if blacks queen was on g5, h3 and d3 (the last two after recapturing on d7 and then moving there). However after some thought it became clear these also required too many tempi. Then I started thinking about the check being on h4 and mating the white king on h1 with a discovered check. Given that white castling saves a couple of tempi this seemed plausable and after a few minutes I found the solution. The answer to the logic puzzle is *[correct solution given]*. I thoroughly enjoyed both puzzles, thank you very much for bringing them to my attention.

**Sylvain Carre, Lausanne, Switzerland**

It took me approx. one hour to find it. I quickly understood that it had to be a discovered or double check. I wanted to mate the king on d2 (Qg5, Ne3 to Nf1) and it did not work. When I realized that the king had time to go to h1 the solution became easy.

**Luke Harmon-Vellotti, Boise, Idaho**

The answer that I found was *[correct solution given]*. I was a bit surprised to find this , I was thinking of ways of moving the knight from h2 to f1 as a joke, but somehow it works, I was also surprised since using this pattern the move order isn't very important so I think there are 272 solutions which is quite a lot for this sort of puzzle.

**Dominic Menard, Canada**

This was a lot of fun: *[correct solution given*]. I was trying to find a solution with my children for half an hour. Then I went to the kitchen to prepare the breakfast. I was wondering, "How could I save a move". I was always missing on. Then it came to me!! Castling!! I went back at the board and found the solution very quickly. So I found the main move without a board. Happy new year! And thanks for the Christmas problems. For the bonus challenge: just use an invisible cloth. Haha, I'll keep working on it.

**Themis Argirakopoulos, Athens, Greece**

Double check against white King was the first idea, but a setting with wKd2, bNe3, bBh6 was not possible to be realized, unless 6...Nxf1# was the final move. Another important point was that both bQ and bB could be used as the back piece in that battery. Ok, there is no unique order in moves in such problems, but a solution with different battery pieces is definitely a big minus! So, a discovered check was the second thought. Combined with a tricky castling and how easy a bQ can be at the h-file and the solution was there. Took me about half an hour.

**Carlos Mas, Monterrey, Mexico**

It took me two nights in bed, while trying to sleep, to solve the problem, Effective time about 30 minutes. First I thought a discovered check was not the solution because the king would take on f1, so I was trying to find a solution with a double check instead. Then I thought of a discovered check when the king is unable to take on f1. All this went to my head without even seeing the board. Sometimes doing that helps both solving chess probelms and falling asleep. About the coins problem, I don't get it, it seems to me that it's impossible to solve it without further clues. Or maybe I just need an additional couple of nights to solve it.

**Clément Sreeves, Edinburgh**

I spent around three hours looking for the solution, with no success. All the obvious squares that the king would get checkmated on failed by one move or one square left uncovered. Frustrated, I decided instead to improve my computer programming skills by coding up the rules of chess in my favorite language, Python. I figured this might allow me to brute force my way to the solution! While figuring out which data structure I should use to represent positions, all of a sudden the solution to the proof game dawned on me. Now I'm left with my programming challenge, which is also proving to be rather tricky!

**Michael Henderson, Eagle, Idaho, USA**

The solution is not unique, although the basic structure of the checkmate is, I guess. I really can't give you the precise time it took me to solve this because I've been busy with other things and I haven't had time to sit down with a chessboard and really look at it. So I just thought about it in my head from time to time. I found lots of tries that fell just short, as I'm sure other solvers did, but I didn't think about castling. So this morning at about 4:00 a.m., as I was being kept awake by the snoring of our pug (one of two dogs who sleep in the bed with us), I thought about it again and it finally dawned on me. "Hey, what about castling!" Once I began to think about positions involving castling I had the solution in about 4 or 5 minutes. So thanks to Maggie the pug.

**Eric Black, Illinois, USA.**

I spent 32 minutes on the problem. I tried to make the White king on d2 for a mate initially. I realized there were not enough moves. I then tried to make the White king on h2. Again not enough moves. I then thought the solution must be a discovered checkmate. I had 4 Black knight moves but only 2 other Black moves to utilize. So I believed the Black queen must go on h4. It then occurred to me that White must castle for this to work! I found the solution right away after that.

**Jason Rihel, London**

I am usually good at these. This one took me several days, an hour or two (!) a day. I had numerous seven move variants. I had a move the knight only variant that would take 6 moves if only the White King could stay in check. I finally had a flash while sitting in a coffeeshop when I randomly realized I had long ago 'dismissed' the correct family of variations for the silliest reason-- Nf3 hits the queen on h4 but could simply be put out of the way on e2! That carelessness cost me hours! Thanks for the puzzle.

**Paul Cooper, Armenia**

*[Correct solution given]* I spent a few hours looking at ideas involving e4 and Bb5xd7 or Qg4xd7. Also e4, Bd3, Qf3, Ke2-e3, Ne2 with Black e5, but always one move over or captures f1 for mate. A good challenging problem.

**Roland Kensdale, Ellon, Scotland, UK**

*[Correct solution given]* I spent a few minutes looking at mates with the knight alone with a white king on d2 or h2, and then realised the solution must be a discovered double check. I found a game ending in 7...Nf1 with white Ke3 and black Q on g5, but it took too many moves to block the white king's flight squares. I looked at a mate with a black R on h3, again white king on e3. I needed one move to move the h pawn (captured by white the w piece must then move) but then couldn't block the king's flight squares with white piece/pawn moves. A black e7-e5 removed enough flight squares but then no way of capturing the black pawn on h7, moving the White piece which captured and also blocking the remaining flight squares around the king. Once I thought of a mate with the king on h1 using castles to get the king to h1, I found the solution very quickly - possibly 5 minutes, but had spent around 2.5 hours on the other attempts.

**Vítor Almeida, Juiz de Fora, Brazil**

I've worked in this problem whenever I had a break. In total I've spent about six hours to resolve. First I was trying to make a mate on h2, g3, e3 or d2, struggling trying to remove the bishop and the queen to make f1 a free square to Black's knight. Didn't work. My focus was on e3, trying to create a trap on center releasing the Black's queen. It was bothering me, so I decided to do without the board, when I would sleep, and suddenly I had the idea to try a discovered check. h1 was my last try and when I saw the solution I yelled out happily and scared my parents. I have about 2000 Elo and I am trying to play chess seriously.

[Event "QM3"] [Site "?"] [Date "2016.01.02"] [Round "?"] [White "Rachels, Stuart"] [Black "A games ends in 6...Nf1#"] [Result "*"] [PlyCount "12"] [EventDate "2016.??.??"] 1. e4 ({Extreme variation:} 1. e3 e6 2. Ba6 Qh4 3. Ne2 Nh6 4. O-O Ng4 5. Kh1 Nxh2 6. Rg1 Nf1# {[%cal Gh2f1]}) 1... e5 2. Bc4 Nf6 3. Ne2 Ng4 4. O-O Qh4 { There are many different ways to reach the position on the bottom right of the board, but all variations converge to it. From here on they are identical, and the key idea we needed to find is shown in the following two moves:} 5. Kh1 Nxh2 6. Rg1 {[%cal Gh2f1] [#]} Nf1# *

*There are 36 coins on a table, ten showing heads and the rest tails. They are covered with a cloth. You must reach under the cloth and separate them into two groups, not necessary equal in number, turning over any and as many coins as you like. You cannot feel or in any way tell whether a coin is showing heads or tails. When you are finished and the cloth is removed both groups should show an equal number of heads. How do you do it?*

I told the story of showing the problem to Vincent Keymer when he met Garry Kasparov in Berlin. During breakfast with the ten-year-old chess prodigy and his father, who teaches music at a German university, I gave them the John Nunn's coin problem and they spent quite a while thinking about it. When we were leaving they hadn't worked it out, so I quickly gave them the five-second solution:

*Separate ten coins and turn them all over.*

As mentioned previously both thought for a while and then Christof Keymer, the father, said: "I don't get it!?" To which his son, whose face had lit up, said something truly inspiring: "Don't worry, dad, I'll explain it to you on the plane back." Sent shivers down my spine. Anyway, you can work out for yourself why the above is the solution – or take a plane trip with a ten-year-old.

*The winner of the special New Year's prize was Peter Finn of High Wycombe, England.*

Discussion and Feedback
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Jochen Keller 1/11/2016 10:02

Addendum:

The reason I only considered unique solutions might be the fact that the previous two proof games also fulfilled this standard. I am quite angry now, for the first time in more than 10 years of CB Christmas puzzles.

Anyway, a happy new year to you all!

The reason I only considered unique solutions might be the fact that the previous two proof games also fulfilled this standard. I am quite angry now, for the first time in more than 10 years of CB Christmas puzzles.

Anyway, a happy new year to you all!

Jochen Keller 1/11/2016 09:56

I feel really close to GM Nunn now, as I tried for many hours to find a unique solution: pawn pushes and captures, bishop sacs on a6 and d7 and so on. I didnt even consider a solution with multiple possibilities.

Please mention that next time when you give the problem, not just when posting the feedback.

Please mention that next time when you give the problem, not just when posting the feedback.

Joshua Green 1/9/2016 05:31

Ah, good point! I'm not sure how I missed those, but now I'm satisfied that 256 is correct.

Lars Rasmussen 1/9/2016 05:27

It looks like Joshua missed the 2 Black sequences with 4. ... Nf6-g4. There are only 2 since they must start with e6/e5, Qh4 and Nf6. 12 + 6 + 12 + 2 = 32, and 32 * 8 = 256.

cashparov1 1/9/2016 04:37

@Joshua Green

I only counted the variations with 4... Nxh2 (I excluded 3... Nxh2 since it makes castling impossible) and I arrived at 256 variations. :-)

I only counted the variations with 4... Nxh2 (I excluded 3... Nxh2 since it makes castling impossible) and I arrived at 256 variations. :-)

Joshua Green 1/9/2016 03:57

I don't think there's an easy way to get the number of solutions to the 6. ... Nf1# problem. The trouble is that the move orders interact in surprising ways. I get 240, and I can see exactly how the previous count of 256 mistakenly counted an extra 16.

To begin with, let's look at White's moves. They must be 1. e3/e4, 2. Bd3/c4/b5/a6, 3. Ne2, 4. 0-0, 5. Kh1, and 6. Rg1. Thus, White has 2 x 4 = 8 possible sequences, and (with one exception) they don't interact with Black's moves.

Now let's consider Black's moves. At some point he must play e6/e5, Nf6/h6, Ng4, Qh4, Nxh2, Nf1#, but these moves get in each other's ways. Specifically:

----- Nf1# must be the last move, of course.

----- Ng4 must be played before Nxh2.

----- e6/e5 must be played before Qh4.

----- Qh4 cannot be played while/if the bN is on f6.

----- If Black plays 3. ... Nxh2, White won't be able to play 4. 0-0! (This is the point that the above solutions overlook.)

To count up these possibilities, I decided to work backwards. Let's retract moves:

---- 6. ... Nh2-f1#

-------- 6. Rf1-g1

------------ 5. ... Qd8-h4

---------------- 5. Kg1-h1

-------------------- 4. ... Ng4xh2

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nh6/Nf6, and Ng4. There are 12 possible orders.

------------ 5. ... Ng4xh2

---------------- 5. Kg1-h1

-------------------- 4. ... Nh6-g4

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nh6, and Qh4. There are 6 possible orders.

-------------------- 4. ... Qd8-h4

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nf6/h6, and Ng4. There are 12 possible orders.

We therefore get 12 + 6 + 12 = 30 sequences for Black. Multiplying by the 8 sequences for White yields 240 total solutions.

To begin with, let's look at White's moves. They must be 1. e3/e4, 2. Bd3/c4/b5/a6, 3. Ne2, 4. 0-0, 5. Kh1, and 6. Rg1. Thus, White has 2 x 4 = 8 possible sequences, and (with one exception) they don't interact with Black's moves.

Now let's consider Black's moves. At some point he must play e6/e5, Nf6/h6, Ng4, Qh4, Nxh2, Nf1#, but these moves get in each other's ways. Specifically:

----- Nf1# must be the last move, of course.

----- Ng4 must be played before Nxh2.

----- e6/e5 must be played before Qh4.

----- Qh4 cannot be played while/if the bN is on f6.

----- If Black plays 3. ... Nxh2, White won't be able to play 4. 0-0! (This is the point that the above solutions overlook.)

To count up these possibilities, I decided to work backwards. Let's retract moves:

---- 6. ... Nh2-f1#

-------- 6. Rf1-g1

------------ 5. ... Qd8-h4

---------------- 5. Kg1-h1

-------------------- 4. ... Ng4xh2

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nh6/Nf6, and Ng4. There are 12 possible orders.

------------ 5. ... Ng4xh2

---------------- 5. Kg1-h1

-------------------- 4. ... Nh6-g4

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nh6, and Qh4. There are 6 possible orders.

-------------------- 4. ... Qd8-h4

------------------------ 4. 0-0

---------------------------- Black's other moves must have been e6/e5, Nf6/h6, and Ng4. There are 12 possible orders.

We therefore get 12 + 6 + 12 = 30 sequences for Black. Multiplying by the 8 sequences for White yields 240 total solutions.

joscho 1/9/2016 03:10

I can also confirm 256. First I ended at 224, but that did not cover all cases of 4. .. Nh2:. I found out by doublechecking with Lars' explanation. Nice "puzzle after the puzzle"... :-)

Thanks to Chessbase and Frederic in particular for making me spend (too) much time on these puzzles.

Thanks to Chessbase and Frederic in particular for making me spend (too) much time on these puzzles.

cashparov1 1/8/2016 09:36

I've also arrived at 256 variations, including those with the knight taking on h2 before the queen comes out to h4.

My methodology was the following:

I first created a pgn file in a computer program with all the possible variations after 1. e4 e5 2. Bd3. I got 7 variations that way. Now since the bishop can also go to c4, b5 and a6, I multiplied my result by 4. So that gives me 7x4=28 variations in the 1. e4 e5 tree.

Now whether the black pawn goes to e5 or e6 on the first move is basically the same thing, so that means the 1. e4 e6 tree is also composed of 28 variations.

Then I checked the 1. e4 Nf6 2. Bd3 tree. That's 8 variations. Multiply that by 4 possible bishop moves and you get 32 variations for the 1. e4 Nf6 tree.

Finally I checked the 1. e4 Nh6 2. Bd3 tree. I got 10 variations multiplied by 4 again which gives me 40 variations for the 1. e4 Nh6 tree.

So that's a total of 28+28+32+40=128 variations for the whole 1. e4 tree.

And since whether white goes e3 or e4 on his first move doesn't alter the position, the whole 1. e3 tree is also 128 variations.

Hence: 128+128=256 total possible variations ending with 6... Nf1 mate. :-)

Did I get that right Mr Nunn? ;-)

My methodology was the following:

I first created a pgn file in a computer program with all the possible variations after 1. e4 e5 2. Bd3. I got 7 variations that way. Now since the bishop can also go to c4, b5 and a6, I multiplied my result by 4. So that gives me 7x4=28 variations in the 1. e4 e5 tree.

Now whether the black pawn goes to e5 or e6 on the first move is basically the same thing, so that means the 1. e4 e6 tree is also composed of 28 variations.

Then I checked the 1. e4 Nf6 2. Bd3 tree. That's 8 variations. Multiply that by 4 possible bishop moves and you get 32 variations for the 1. e4 Nf6 tree.

Finally I checked the 1. e4 Nh6 2. Bd3 tree. I got 10 variations multiplied by 4 again which gives me 40 variations for the 1. e4 Nh6 tree.

So that's a total of 28+28+32+40=128 variations for the whole 1. e4 tree.

And since whether white goes e3 or e4 on his first move doesn't alter the position, the whole 1. e3 tree is also 128 variations.

Hence: 128+128=256 total possible variations ending with 6... Nf1 mate. :-)

Did I get that right Mr Nunn? ;-)

Lars Rasmussen 1/8/2016 07:31

I think the correct number of variations is 256. The number 160 does not take into account that Black can take on h2 before bringing the queen out.

Assume Black plays e5 and brings the knight out via h6. Then Black's first 5 moves consists of 2 moves by pawn and queen and 3 knight moves. 2 out of 5 makes for 10 possible combinations, but the one with Nxh2 on move 3 makes castling illegal, so that leaves 9 legal possibilities.

Now if Black brings the knight out via f6 there is only 7 possibilities because the queen cannot jump over a knight on f6.

9 + 7 = 16. Multiply by 2 for Black's option of e6/e5 and by 8 for White's different options. 16 * 2 * 8 = 256.

Assume Black plays e5 and brings the knight out via h6. Then Black's first 5 moves consists of 2 moves by pawn and queen and 3 knight moves. 2 out of 5 makes for 10 possible combinations, but the one with Nxh2 on move 3 makes castling illegal, so that leaves 9 legal possibilities.

Now if Black brings the knight out via f6 there is only 7 possibilities because the queen cannot jump over a knight on f6.

9 + 7 = 16. Multiply by 2 for Black's option of e6/e5 and by 8 for White's different options. 16 * 2 * 8 = 256.

cythlord 1/8/2016 06:09

@Cashparov1 That is totally correct, but what is even more interesting is the rather subtle hint that is given to us. It's pure semantics, but "10 heads and the 'rest' tails," which implies that the amount of tails does not matter.

I'm curious as to what thought process readers of chessbase chose to find the solution (I googled the solution) because, while once given the solution it is painfully clear, it is to me unclear how one arrived at the said solution.

I'm curious as to what thought process readers of chessbase chose to find the solution (I googled the solution) because, while once given the solution it is painfully clear, it is to me unclear how one arrived at the said solution.

darth janis 1/8/2016 03:07

I believe there are 160 possible Variations leading to 6. ... Nf1#

White has to start with e3 or e4 and follow it up with one of the 4 possible moves by the Bishop. So for every Black move order there are 8 ways to Play for White.

Black can start with either Nf6 e6 e5 or Nh6

After Nf6: there are 4 ways: e6/e5 has to be played now or after Ng4

After Nh6: there are 6 ways: because after e6/e5 black has the Option to start with Qh4 and only then Play Ng4

Aftere5/e6: there are 5 ways each, Qh4 Nh6 or Nf6 on move two and Nh6 and Qh4 allow for one more choice at move 3.

So there are 2*5+4+6 ways to play for Black and for every Black moveorder there are 8 ways for White

(2*5+4+6)*8=160

White has to start with e3 or e4 and follow it up with one of the 4 possible moves by the Bishop. So for every Black move order there are 8 ways to Play for White.

Black can start with either Nf6 e6 e5 or Nh6

After Nf6: there are 4 ways: e6/e5 has to be played now or after Ng4

After Nh6: there are 6 ways: because after e6/e5 black has the Option to start with Qh4 and only then Play Ng4

Aftere5/e6: there are 5 ways each, Qh4 Nh6 or Nf6 on move two and Nh6 and Qh4 allow for one more choice at move 3.

So there are 2*5+4+6 ways to play for Black and for every Black moveorder there are 8 ways for White

(2*5+4+6)*8=160

Odomok 1/8/2016 01:42

Of comments I read was "I was also surprised since using this pattern the move order isn't very important so I think there are 272 solutions which is quite a lot for this sort of puzzle". A nice puzzle is to determine how many solutuons exist, I think it is more than 272..

Hawkman 1/8/2016 11:01

"A part two report with the prize winners, norm winners and some exclusive pictures from the closing ceremony will follow shortly."

That was posted in the final full Qatar report a week ago, but it's never been published here. What happened? Where is it? Will it get published at all someday?

That was posted in the final full Qatar report a week ago, but it's never been published here. What happened? Where is it? Will it get published at all someday?

cashparov1 1/8/2016 10:51

what is fun about the coins problem is that the solution works for any number of coins. if you have say 1976 coins on the table, 666 are heads and the remaining 1310 are tails, you separate 666 coins, flip those 666 coins over, and presto, you have two groups with an equal number of heads. :-)

but don't take my word for it. try it out with 1976 coins. ;-)

but don't take my word for it. try it out with 1976 coins. ;-)

sicilian_D 1/8/2016 07:48

Happy to see some of us readers being brought into the main reporting area of ChessBase. (I am included as well).

Thank you ChessBase, and Mr. Frederic Friedel

Thank you ChessBase, and Mr. Frederic Friedel

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