The game R.Byrne - Fischer, U.S. Championship 1963 (see that post for background and a link to the first 21 Move Brilliancy in this series) is one of Fischer's most famous wins, partly because of the strength of Fischer's opponent, himself one of the leading American players; partly because of the sudden end (Byrne: 'at the very moment at which I resigned, both grandmasters who were commenting on the play for the spectators in a separate room believed that I had a won game!'); and partly because people often confuse it with the 'Game of the Century', Fischer's even more famous 1956 win over Robert's brother Donald.
As I pointed out in The Longest Analyses, Fischer devoted a full page to White's 14.Rfd1, calling it 'another of those melancholy case histories entitled "the wrong Rook"' and suggesting 14.Rad1, an assessment with which all subsequent analysts agree, Kasparov included. [Re "the wrong Rook", what other games are well known?] After 14...Nd3, the game reached the position shown in the diagram.
U.S. Championship 1963
Byrne played 15.Qc2, where Fischer commented,'There is hardly any other defense to the threat of 15...Ne4', and among other tries gave '15.Nd4 Ne4 16.Nxe4 dxe4 17.Bb2 Rc8 with a powerful bind'. Kasparov gave 15.Qc2 a '?' and commented, 'This loses the game, but it is not easy to find a more tenacious defense. The following variations are possible:', where he built on Fischer's analysis. After repeating the entire variation from 15.Nd4!? through 'powerful bind', he added, 'but Huebner rightly considers this to be White's best chance and he suggests 18.a4. However, here too after 18...Qg5 Black has an obvious advantage thanks to his powerful Knight at d3.'
In An Olympiad Bind, I already discussed the difficulties of using an engine to analyze a bind and won't repeat those points here. Kasparov might agree, because, unlike the Olympiad example, he gives no further analysis. There are alternatives to both 18.a4 and 18...Qg5, and the best continuation is not obvious. The computer wants to swap a pair of minor pieces on b2 and another pair on d4, then go after White's isolated Pawn on d4. A GM might prefer to torture White by leaving the minor pieces in play and building pressure with the Rooks on the c-file.
Whichever move loses the game, 14.Rfd1 or 15.Qc2, both moves were necessary to arrive at a hopeless position on move 21. It also took Fischer's genius to find the startling continuation that crushed his opponent. With this kind of form, it's little wonder that he finished the event with a perfect +11-0=0, a preview of his match whitewashes in 1971.