WORKSHOP (2) - PROBLEM ONLINE TT1

by Itamar Faybish

Well, I have so many people to thank, but I already did most of it in the comments part. Just once again, thanks to Problem Online for staging this competition in a very professional manner. And a (very) special thanks to Ivan Bender for his kind and generous spirit.

This part will be a more extensive analysis of the competition, and more particularly the main variant. Actually we will not dwell on the other ones. I will present mostly in an informal style, and I hope you will not fall asleep before reading at least half of it... :-)

Well, how did I ever end up in this one?! Those who know me, know that I love challenges, any challenge, give it to me and I probably won't sleep until it's solved... :-) You know now how to torture me, but please, please... keep it a secret...

Every now and then there is a challenge which takes more time than usual. For example, before the TT1, my challenge was to solve the connect-4 game. Those of you may know this game, the rules are very very simple, you have a 6x7 rectangular board, each player has 21 discs which they put one by one in a column of their choice, at the lowest possible point, each turn. The goal being to make a 4 consecutive discs pattern.

Solving it means to find out who wins, and with what first move. That was quite a tough one, but slowly by slowly I got my search algorithm better, until it was solved. I was so happy, a - Happy men -... :-) I wanted to "dispatch" this information to the "world", so naive was I... as a matter of conscience, I looked if someone had not already solved it... Mmm, it was quite funny, yes, I found that it was solved previously, in a Master thesis, in 1988 (!!!), by Victor Allis! hehehe :-)

Well, I did not regret working on it, as it was quite fun, with an interesting ending. But I should have looked for the literature about it first, a lala...

Then there was some calm, before I came across the Problem Online TT1. Well, truthfully, at first, it seemed strange, I never encountered this sort of problem. Some time passed and vacation came. I must say that what happened then was unexpected. Suddenly I recalled this competition, and I went to have a look again, just to see a little bit more, and who knows, I could maybe just - participate - ?!

Then I started composing, and I got some nice leads, good Itamar! you ARE getting somewhere (auto laugh...). But, it got over me, and slowly by slowly, I was hooked. Then it was just impossible for me to do anything but to try and solve this. I went from the start for the 37 squares area, maybe that was not such a good idea, but luckily I advanced.

I must say that such problem is very tough when one does not really know where one is going. I did not know if a solution existed in the number of moves I looked at. It is most probably just sheer luck that I succeeded. Actually I could resume the needed factors: luck, strategy, logic, and determination. All of these are needed.

Without luck, even with a promising line, one could spend months, if a solution does not exist, it simply does not exist... strategy, well that means how to look for solutions, in what manner. Which lines, even promising, should be discarded, and which ones to continue working on. Logic, well, these problems DO need some sense of logic... :-) And finally determination. Well, I believe this one is actually even much more important than the previous three.

I am a firm believer in determination. When one is fully determined for something, one usually succeed, doesn't matter the walk of life. I think it is Thomas Edison who first said something like: “Genius is one percent inspiration and ninety-nine percent perspiration.” I a hundred percent agree... :-)

Thus I was determined, that much is clear. But can you imagine? I took a week vacation, for, well, vacation... how many of you spend vacations solving a problem??? :-) Well, maybe some of you... I do not normally spend those on these, but then again, destiny has it differently.

And, well, of course I do not regret working on it. This is one of the best challenges of my life, beside, problem solving is my business... :-)

When I sent the solution the first time, I told Problem Online that it would be interesting if I could write an article explaining how I got this solution, from a technical point of view. That was a good idea, and Ivan Bender agreed, but then, the timing was not good.

The competition was far from over, and many things could change, like a better solution found. I thought at the time that a better solution did exist, although it would be very tough to find. I looked a little bit for a 13.0 moves solution, but it looked almost impossible to make a complete tour.

But a 13.5 moves with less knight takes DID look possible, if not even probable. Am I not coming from nowhere with my solution, having not much experience in chess problem solving, with all of those experts out there?! Mmm, thinking that mine was the best would just prove that I have a big ego... :-) No, I really thought someone would overtake me at some time, and there was a lot of it until the 12th of august...

Thus such document explaining in a technical manner was not made at that moment. I would say unfortunately, as at that time I was really "high". My mind was seeing problems everywhere... :-) And everything was fresh, all my impressions and all. Ok, now I write about it, and I will try to recall what I thought at that moment, hopefully it will not be too bad.

Thus, "where shall we begin" I thought at that time. We have a chess board, and we have a knight tour to make.

A, I just looked at the history of the tries, I get nostalgic... :-) Here is my first try, please, you can laugh your heart out, no problem... :-)

1.c3 b5 2.Qc2 b4 3.Qxh7 Rxh7 4.Na3 Rxh2 5.Nb5 Rxh1 6.Nxa7 Rxg1 7.Nxc8 Rxa2 8.Nxe7 Rxb2 9.Nxg8 Qg5 10.Nh6 Qxg2 11.Ng4 Qh2 12.Nxh2 Rxf1+ 13.Nxf1 Rxd2 14.Nxd2 b3 15.Nb1

Yes, I thought THAT might have a chance of being correct, now I would discard it without ANY attention... :-) Just crazy, take take take, and more take. Even if by unbelievable luck it ends up being correct, it would take ages to be proved!

Ok, not a very good try, but a good beginning nonetheless. The idea 1.c3 b5, followed by Qc2, something, Queen something, followed by the start of the tour looks very promising! I rapidly improved that (well, actually not rapidly, there is a strange gap between the 22/03/2005 and 25/03/2005, I wonder what I did during those days, apparently something very important as I did not make a singly try during this period! Maybe I just slept...)

The idea above seems promising. And maybe it can give a good solution with more thinking. But some improvements are needed, for example 3. Qxh7 seems quite silly, why not play 3. Qg6 hxg6 ? Or maybe instead of Qc2, Qa4 followed by Qc6, then the d or b pawn takes the queen. Yes, looks a little bit better. Also 1. ... b5 2. ... b4 releases the bishop on c8, which diminish the probability of it being correct. Thus:

1.c3 Nc6 2.Qc2 Ne5 3.Qg6 hxg6 4.Na3 Rxh2 5.Nb5 Rxg2 6.Nxa7 Rxf2 7.Nxc8 Rxa2 8.Nxe7 Rxb2 9.Nxg8 Qh4 10.Nh6 Qxh1 11.Ng4 Qh2 12.Nxh2 Rxf1+ 13.Nxf1

was tested. But still, I rapidly found out several points, they may not be correct all the time, but maybe good general points that experienced solvers already know (they could be plain wrong that is, please have mercy, I'm no expert... :-) ):

- The MORE pieces one takes, the LESS probable it is for the solution to be correct. This might seem paradoxical, or maybe not. The problem with taking more pieces is that the "phantom" of these pieces can roam the board wherever they want during the game! That said, maybe a more precise statement needs the word: inappropriate after the MORE. For example taking a bishop which is blocked by his pawns on the initial squares, with a knight (or other) may actually prove to be very good. But the statement is of a general nature.

- One piece that one does not want to take is the Queen, unless in special circumstances.

- With respect to the first point, the more pieces one takes, the more time it takes Euclide and Natch to investigate the position! And when I say more, it may well be exponential! Just taking one pawn more in one solution than another can have an enormous significance!

- It is very good when one takes a piece which cannot move because obstructed by his own pawns.

- It is best to have a very clear logic from start to finish, although one may get away with it in many circumstances. If a move is directly dependent on the previous one made, it has more chances of being correct.

- Having a (or many) promoted piece(s) with the other equivalent piece(s) all still on the board is very good! It forces the solution to have a promotion.

- It is best if the pieces make moves which take forced tempos, for example if a king moves, then he should not come back to the same square but maybe continue on its way, making a difference of two squares with the original one. Thus the solution needs to incorporate those moves.

Ok, enough blabla, after those tries, I started to look at another promising line

1.c3 Na6 2.Qa4 Nc5 3.Qc6 dxc6 4.Na3 Bh3 5.Nb5 Bxg2 6.Nxa7 Bxf1 7.Nc8 Rxa2 8.Nxe7 Rxb2 9.Nxg8 Bd6 10.Nh6 Bxh2 11.Ng4 O-O 12.Nxh2 Kh8 13.Nxf1 Qxd2+ 14.Nxd2 Rd8 15.Nb1

This seems already closer to something which might be correct. That said, it is not one I would test now, for example 7. ... Rxa2 8. ... Rxb2 are not necessary. This one gets demolished at the 13.5 stage... By the way, I saw soooo many "Problème démoli" by Euclide, argh... but then again, this is more than fully compensated by the few but o so nice "Solution Unique"... :-)

My Last try with this scheme was on the Saturday 26/03/2005. Then, at 23:35 that day, I switched to a 13.5 moves solution promising line. And finally, at 03:18 on Sunday, I found the solution, which at that time I did not know was ok, but after a few hours of Euclide's work, it turns out well. I cannot say how happy I was at that precise moment... :-)

Till now I talked a lot, but not much analysis was presented...

Let's analyze the final solution:

1.e4 d5 2.Bc4 dxc4 3.Na3 c3 4.Nb5 cxb2 5.Nxa7 b1=R 6.Nxc8 Rb3 7.Nxe7 Rf3 8.Nxg8 Bd6 9.Nh6 Bxh2 10.Ng4 O-O 11.Nxh2 Kh8 12.Nf1 Qxd2+ 13.Nxd2 Rd8 14.Nb1

Ok, the first moves are capital. We need to get the knight going all around, how can we do that?! Are we going to make a tour with a black knight or with a white knight? Also, which knight shall we choose, the b1 or the g1 knight?

This last question is very important, and very subtle. We know the routes the knights have to take, so why choose one knight instead of the other as a more "promising" knight?

Well, the answer comes mainly from the two squares g8 and b8. Can you see the difference? Once the b1 knight gets to g8, it has then g4 as a destination. With the black king at e8, the knight can only pass by h6, unless the king moves of course, but we assume it does not at that stage. Considering the other knight tour, when the g1 knight gets to b8, it has a6 and c6 as possible destination for the destined b4 square.

This might seem a little bit too "simple". And actually, well, in a way it is... :-) The b1 knight does seem more promising as such, but the g1 knight tour IS interesting! I passed some time analyzing it and found some reasons to believe a 13.5 correct knight tour might exist with it also.

Now, can we make better than 13.5 moves? It is trivial to see that the best solution has to be greater than 11.5 moves. 12.0 moves is out of the question as nothing can force a black knight from starting at a6/h6 before moving to b4/g4.

How about 12.5 moves? Well, it is not possible for the same reason. The 13.0 moves?! A, that one is tough... The first moves must be either a white bishop sacrifice on a6 or h6, followed by the black knight's tour, or a white bishop takes pawn b5 / g5 then goes to c6 / f6, etc.

As proved by Göran Wicklund and Miodrag Mladenovic (they found 34 squares area in 13.0 moves! Very nice!), it is not clear that the maximum cannot be achieved in that number of moves. I looked for one but couldn't find.

Then how about 13.5 moves?! As we advance in the number of moves, more possibilities are offered of course. This is the first barrier where a correct solution is actually a promising prospect.

How can we start the white's knight tour? Well, one way is for a black bishop to sacrifice himself at a3 or h3. This gives many many possibilities, but the absence of that bishop which seems necessary for dealing with the h2 / a2 pawn, plus the absence of the e7 / d7 pawn from the start makes it very tough to advance successfully.

Thus it is not very promising - possible, yes - but not promising.

Then what? Well, not many choices, it has to be that the pawn e2 or d2 moves, then black plays his b7/d7 pawn or e7/f7 pawn respectively. Then white gives up his bishop c1/f1, and finally the knight starts its tour. This is VERY promising. Why? Well, once a black pawn takes a bishop, it advances and promotes. As seen previously, promoting to a piece which is already on board increases one's probability that the SPG is unique.

What's more, this promoted piece can play an important role in the course of the game.

Ok, as talked about previously, the b1 knight was chosen as it seemed the most appropriate, but the other one was analyzed and could be correct. Now, where to play on the very first move, pawn e2 to e3 or e4? At that stage, nothing can really be said, more analysis is needed. Then which black pawn to play on the first move? We can choose either the b7 or the d7 pawn. Both were analyzed, but the d7 pawn seemed more appropriate because of the final possible queen sacrifice on d2 preparation.

Then until the 5.0 move, it all seems somewhat mandatory. We have now a choice of what to promote to, and even to take a piece or not. We have to look a little bit further in the play, and ask ourselves how could the white knight finish its tour, and what are the obstacles. Well, one, big obstacle is that when the knight gets to h2, it can either pass by f3 or f1 for his next destination, d2.

How can we prevent that? One way is to make the knight on h2 take a black piece on f1. Another way is to block the f3 square with one of our pieces. What better suited for this task, than our newly promoted piece? Thus, we promote to a rook. This also settles the first move, we will play e4 to let the rook pass.

That said, it is here where interesting possibilities might lie. For example promoting to a knight, then taking the d2 and h2 pawns with it, or some other lines. I looked at many such lines but did not find much.

Until the 8.0 moves, everything falls into place, perfect timing. Our rook blocks the future arrived h2 knight from going to f3, and now the black knight just took a pawn blocking the black bishop, and released him from his chains.

What's more, this bishop seems very appropriate for sacrificing himself on the h2 square.

And so, this bishop takes the h2 pawn, and the knight comes to grab him. As said before, this knight cannot go there through the f6 square because of the king, contrary to a b1 knight tour solution.

Then when the knight gets to g4, it's a most perfect time to castle... :-) This leaves some more possibilities for continuing the SPG after black moves.

And finally, when the knight reaches f1, the queen sacrifice herself, leaving no choice for white.

Can we improve that? Well, how can we make a 5 knights take tour? One promising way seems to let the knight returns to b1 through a vacant d2 square. But then an unexpected solution appears. We see that once the white knight is on the g8 square, it can reach the b1 route in 4 moves through g8-e7-d5-c3-b1 (or another route if the black king allows). Notice that if the knight made the all round route, g8-h6-g4-h2-f1-d2-b1, it takes him 6 moves. Thus white has saved two moves, which he can use for example to play Rh1x[a piece, for example on h2], Rh1.

This last analysis demolishes many "promising" lines having a 5 knights take.

Well, hopefully you are still awake and kicking... :-)

So, this ends our analysis. What more can be said? Well, now basically two questions remain: first, is there a completely different solution in 6 knights take? Second, is there a better solution?!

François, we are counting on you for this one... :-)

Then, there are other interesting points. Like François's idea for a variant based on the length of the polygon instead of the area.

Before the end of the competition, I thought about a variant searching for the biggest polygon of 0 knights takes. I was amazed seeing that Paul actually came up with a 5 squares area in 0 knights take!

At some time I also thought about the following: the biggest area of a double knight tour, that is, the knight must make twice exactly the same tour successively.

Or even a bishop - or another piece - tour, with the same conditions.

Well we can go on and on... :-) This theme is quite rich!