Hi! Long time no see!_syLph_ nice to meet you!
I got interested in dots and boxes because of Conway-Berkelamp-Guy's Winning Ways. I can proudly tell you that I have beaten John Conway at dots and boxes, and when I had a meal with Berlekamp in Berkeley whilst I was supposed to be doing number theory, he declined to play me on 5x5, saying that he knew he would be hammered. They were both a lot older than I was.
The reasons I was good at dots and boxes are easy to explain. First, as a mathematician I had understood the theory of chains and nim, and during my early studies of the game I made much use of a C++ program by Glenn C. Rhoads and Freddy Y.C. Mang which computed nim values of positions, which I used to get a "feeling" for this invariant of a position. When I had become an expert at this sort of thing (knowing e.g. the cunning move in a 2x3 corner with nim-value 4 and all this other high-falutin' nimber stuff) I started playing here and I was getting hammered off the board by 16 year olds who knew absolutely nothing about nim because they realised that it didn't matter what the nim-value was, you actually have to count the endgame to know whether or not you should lose control. Keeping control costs money in the real world, in contrast to the abstract world of nim. The authors of Winning Ways have these throw-away statements that "in our experience the best way to play is to make sure you win the nim battle even it means sacrificing a piece or two" or whatever they say -- "your best odds are in the chain battle" or somehing, but this approach had not dated well, and the players at littlegolem and jijbent were well aware of this. They make a couple of 4-loops (something I would explicitly aim for as P2 in order to complicate the game -- as I got better and realised it was probably a P1 win because my P1 win ratio was so high -- note that 3x3 is a 6-3 P2 win IIRC but I am now very rusty on these matters). I realised that after about 20 or maybe a few more moves I was in a really good position to be able to count the position on pencil and paper and win, because I had proved theorems precisely classifying the cost of a 1x1 and a 2x1 corner using a more refined metric than the nimstring one, and this result about how you can add long loops: 4+6=5 in the sense that these games play in exactly the same way -- this is a stronger invariant than the nimstring one, which just says they both have nim-value 0. I guess Berkelemp did the same kind of thing with go, it's just that dots and boxes is a shorter game, so an important endgame strategy (i.e. counting) has a higher payoff -- I could become an expert dots and boxes player, but he could not become an expert go player until the endgame, by which time he had long lost against the experts.
Every well-played game on this website was 13-12, I remember losing something like 17-8 to Flipster and once feel very humiliated (people often resigned before the end but experts all knew what the final score would be, we were all learning to count). Flipster became better at counting than I did -- I made some conjectures and he proved them for a school project he was working on; we ended up writing a joint paper on the subject which was published in the mathematical journal "Integers": it's article G8 on the Integers website . I was very proud of this paper because we had beaten Conway at his own game, we had proved a much harder theorem than the main theorem of Winning Ways about actual values of games (i.e. beyond nimstring -- for nimstring they were the masters). We had an algorithm which humans could apply using pencil and paper. Flipster was so good at running this algorithm, so young, even before we'd proved it worked, that when I realised that he was playing essentially perfectly from move 16 or so, I accused him of cheating; after a heated discussion I realised that actually he was just extremely smart and I apologised. Cheating, which was always present on the site (in my opinion), is using a computer to analyse a game in progress. This is something I never did, even in high pressure games, although the moment an interesting game I was involved in would finish on a site I would instantly feed it to the dots evaluation program -- I never used the nim evaluation program any more. The dots evaluation program was available online for free, it just analysed the true value of the game by brute force, and there were a few people who would use it to cheat.
The main reason I quit dot and boxes was that the computers were coming. Computers had solved 4x4 long ago (the version played on Yahoo, and the reason I never played there). The computer program Dabble got quite good at 5x5 and the number of clear cheaters on this site began to increase. Dabble was fairly strong, although I could usually beat it even as P2, as long as I was careful and played carefully from about move 15. But I didn't play it much because I just found playing computers boring. My favourite pastime was discovering I was going to lose 13-12 to a human opponent and finding a complex line which I thought would be very difficult for them to play correctly, and then proudly telling them "your mistake was on move 17 where it's essential that you make this sacrifice" at the end. Bill Fraser was a student of Berlekamp and he was working hard on brute-forcing 5x5 for academic reasons, and he let me play on a prototype which had solved every game which had a line from each of the four corner "dots". He turned this into a prototype game which I played privately against a few times: you could choose to be P1 or P2, and the computer's first <= 4 moves were all joining a line from a corner which previously had no lines coming from it (and avoiding some stupid trick where you can lose a box by being dumb) and then after that you were guaranteed that it was playing perfectly, so if you're P1 and not 13-12 up at this time then you're already dead, and you were going to have to work *hard* to beat this thing as either P1 or P2. I knew what was coming, so I quit. I had my publication.
My interest in games then led me to the book by Hearne and Demaine. I got interested in P v NP, whilst trying to understand if there was a better approach than brute force to dots and boxes, and even gave a talk about P v NP at the Royal Institution of Great Britain! https://www.youtube.com/watch?v=A6J9p4iOr3A .
So it's clear that computers can solve dots and boxes -- my (former) hobby. But can they solve my first true academic love -- pure mathematics? This is what I've been trying to figure out for the last few years. I have lost track of the dots and boxes wetpaint blog, but I have a new blog now at xenaproject.wordpress.com where I talk about trying to teach pure mathematics to computers. I never play dots and boxes any more! But with proving mathematical theorems, I am now on the side of computers.
Kevin Buzzard a.k.a. wccanard.