Remember the second match between Deep(er) Blue and Garry Kasparov? This is the one where computers supposedly proved their superiority to human beings in chess. Kasparov had defeated its predecessor the year before by a relatively comfortable 4-2 score, but the rematch in 1997 went differently. Kasparov opened with a comfortable win, but then lost game 2 in what turned out to be controversial style, drew the next three games (though coming close to victory in games 4 and 5), but then had a complete breakdown in the last game, getting crushed in just 19 moves. Thus Kasparov lost the match 3.5-2.5, and the popular media widely broadcast that this was the end of humanity's reign over the royal game. This was almost surely false, but like the old joke goes, we're just haggling over the price; there's absolutely no doubt at this point that decent programs running on decent home computers are far stronger than even the best humans on their best day.
But back to the match. Game 2 was a huge turning point, and not just because the computer won. Kasparov was playing the Closed Ruy - not exactly one of his main openings with the black pieces - and was getting impressively outplayed. A first move that shook up Kasparov was 37.Be4. To human eyes, it's really obvious. Black has no counterplay anywhere, but would if he could play ...e4. Prevent it, and the world is White's oyster. The problem was that 37.Qb6 seems like a simple win of material. The standard view of computers was that they would take free material in all but the most obvious cases where doing so would be wrong, but here it didn't.
But if that bothered Kasparov a bit after the game, that was nothing compared with what happened later. The computer apparently continued rolling along with its positional masterpiece, and when Deep Blue played 45.Ra6 Kasparov threw in the towel. Anyway, analysis later that night seemed to show that in the final position Kasparov missed a draw with 45...Qe3, with an inevitable perpetual check. As the line wasn't even too terribly long, especially for a computer calculating 300 million positions per second, it really seemed to Kasparov that the hand of human intervention was at work. If 37.Be4 was the sort of move where typically human judgment is better than that of a materialistic computer's, then 44.Kf1 is a moment where human judgment falls short relative to the computer's. The engine can work things out to the bloody end; humans can't.
And yet...Deep Blue didn't. How was this possible? After and even during the match, Kasparov requested - or rather, demanded - the logs displaying the computer's evaluation. (Unless I'm badly mistaken, the Deep Blue team claims that they were given shortly after the match, as promised (and even published publicly), but for some reason I haven't been able to ascertain Kasparov continued to insist on their release.) Logs or no logs, it still seemed surprising that Deep Blue missed the perpetual.
But did it? There has been a lot of analysis of the ending since then, and if anything has become clear it's that if there is a draw, it isn't a trivial one. Whether it's a draw at all is an interesting question in its own right, but it's a very different issue from that of the alleged perpetual check. Also, importantly, this non-perpetual line is also relatively obvious for the computer; that is, it's plausible and well within its search horizon. It's clear that White - Deep Blue - is better in the resulting position, but this doesn't settle all the doubts either.
Did Deep Blue spot the perpetual check variations? IF it did, then it would have had to assess the non-perpetual variations as better for it than the endgame after 44.Kh1 Rb8 45.Qc6 Qxc6 46.dxc6. Contemporary engines prefer 44.Kh1, but did Deep Blue?
I've culled material from various Web sources: Ken Regan recently addressed the topic, Wikipedia has a decent introductory survey on the issue, and there are various analytical suggestions in the thread devoted to the game on ChessGames.com. For a historical look back readers may wish a look at this ChessBase article. Finally, I've amassed the analysis from the listed sites and added some of my own here, which you can replay and download. Doubtless there are mistakes in my analysis - I ran it on an older computer, for one thing - so I hope that those of you with access to souped-up hardware will find some improvements.