15.08.2005. Itamar Faybish, Belgium

This competition was quite special and I am happy to have participated in it. Thank you a thousand times for it!

All its participants should also be "praised" in a way. Finding SPG's satisfying the conditions is not easy, and sometimes take a lot of time and energy, with a lot of frustrations when the solutions are so near, yet so far - some may understand -... :-)

I will not write a lot in these comments as it could fill a book, and probably bore everyone to death, just some notes about each of the variants:

main: well, the first one I came across, and one that took me a week to solve, days and nights included... :-) Yes, during my week vacation... It is not an easy one, I first got distracted by a 14.5 promising line, until I found the correct path.

Once the first few moves were found and seemed promising, the solution was not far. Although to tell an anecdote, I began to be really frustrated on the last day of the week, Sunday evening. I was on the point of quitting and left Euclide (thanks a thousand times Etienne Dupuis for this application, and Pascal Wassong for Natch/iNatch!), checking one of my last tries. And in the morning, it was ecstasy... :-) Yes, without it being correct, I probably would have left the competition.

variant A: well, just after sending the main solution, problem online decided to create two new variants... :-) Mmm... I tell myself, Noooooooooo... hehehe... ok, the shock past, it was actually one of the best things they could have done, I liked variant A, less variant B. I thought one knight is already trouble, two is mega trouble.

It was a very tough variant and I also spent quite a lot of time on it, although as my vacation was over, I had to pass nights on it. An interesting point is that, as many competitions prove - quite trivial actually - -> competitors are good for the competition! When one finds a solution, the others try either to equal that, or to do better. When there are no incentives to search, one waits mostly.

Luckily for me, actually I did not wait, for the last ones. As soon as I found the 24-14.5-4, I looked for the 28 area, and rightly so, Göran Forslund found an incredible: 24-14.5-3 soon afterward. That is very tough and Göran found one, and a beautiful SPG (the idea of check with the Queen on h4 then Nf2 followed by Queen h3 is very nice).

variant B: a, this is a story in itself but maybe too long for here thus I will summarize. Well, Christoph Fieberg found a solution if I remember well, relatively soon after the variant was published. I, and I guess others too, were quite surprised at the values. They seemed impossible to beat, all one could hope is to find how he did it.

I did not look much at it at the beginning as for the few times I did, I got to a dead end. I got results that were faaaaar far away than what Christoph found, thus, as the saying goes: let it be. But it stayed somewhat at the back of my mind. Then I left somewhat the competition until the variant C came, which encouraged me to look again at variant B.

Then it was really like a Sherlock Holmes detective story, each time finding some new clues pointing to the answer, until it became evident, one day when waking up. I did not have to go and check with Euclide, it HAD to be the solution... :-)

It is really a special variant, quite a unique and beautiful one, just look at the answer, looks so "simple" yet it works! Given the solution, I looked for improvement that seems to exist but could not find.

Christoph should receive all the praise for finding it so fast, and being the first. For me it was more like solving a math problem where the input/clues are given. Not easy, yet, for Christoph, even the input data was unknown!

variant C: a, that is a very tough one, I was so happy when I first found my miserable 8-15.0-5... :-) but then Paul Raican apparently turned "inspiration mode" on, and produced three SPG, the best being 15-19.0-4, and one incredible 9-17.0-2. How on earth - I thought - did he find two knights tours with a total of just 2 knights takes?!

I found one sequence giving white a 5 squares area with just 1 knight take, then black must make a 4 squares with his knight. I thought naively that this must be how he did it. Well, I can now say affirmatively that this is NOT how he did it... :-) He found an incredible SPG, and apparently the only one in the competition, having a knight tour with 0 takes!!! Incredible! He should be praised for that, it is not easy!

I then found a promising sequence that gave me a 14 squares area. Then I looked either for a 15, 16 or 18 squares area, I found many promising schemes, but could not find a correct solution.

Then Michel Caillaud participated and found a 16 squares area! I was curious to see his game, and I was pleasantly surprised, very nice indeed, very logical!

As an end note, I would say that this competition revealed many secrets offered by this original idea by Ivan Bender (thank you!). Some of these variants may well become classics, not only in the chess world, but in the mathematics world, or even problem/puzzles world. I am sure many mathematicians/logicians would be most interested in it.

Also, these variants are far from being drained, many things are yet to be found for all of them, and improved (well, maybe except the main variant, see François Labelle's comments). And new variants can no doubt be thought of! :-)

15.08.2005. François Labelle, Canada

I used a computer to search directly for the maximal polygon. An exhaustive search showed that 13.0 moves (or less) is impossible. The analysis for 13.5 moves is much harder, so I only analysed the b1 knight and I eliminated some lines that took too long to verify. Considering my analysis and the result of the competition, it's likely that the winning problem is essentially the only way to do a maximal polygon in 13.5 moves.

The progress was extremely slow. When the analysis reached ply 22 (out of 27) I told Alexandre Leroux that the opening that the computer seemed to "prefer" was 1. e{3,4} {b,d}5 2. Bc4 {b,d}xc4 3. Na3 c3 4. Nb5 cxb2 5. Nxa7 followed by a promotion. He completed the proof game in a few days. It took months for the computer to do the same.

Thanks for the nice theme! It is interesting to know that the maximum area polygon is possible. Maybe asking for the longest polygon instead of the polygon with the greatest area would have left a bigger margin for improvement even after have many entries from composers.