The Soviet team consisted of Mikhail Botvinnik, Tigran Petrosian, Boris Spassky, Paul Keres, Efim Geller and Mikhail Tahl. Tahl ended up in the hospital with kidney problems and so was forced to drop out.
On the US team, Reshevsky didn't want to be in Fischer's shadow, and because he couldn't play first board, refused his place on the team which was comprised of Bobby Fischer, Pal Benko, Larry Evans, Edmar Mednis, Robert Byrne and Donald Byrne.
As expected, the Soviets won the finals by a huge margin followed by Yugoslavia (Gligoric, Trifunovic, Matanovic, Ivkov, Parma and Minic), Argentina (Najdorf, Julio Bolbochan, Panno, Sanguineti, Panno and Foguelman) with the United States finishing a disappointing fourth.
The final round was a bitter disappointment for the US. Going in, they seemed a safe bet to medal, but Argentina had been on a roll: they had wiped out Austria 4-0, then beat Romania 3-1. Then in the final round, they crushed Holland by 3-1.
In the last round the US was playing Yugoslavia and a 3-1 win would gave them the silver medal while a 2-2 score meant a bronze medal. But, after two quick draws, a loss by Robert Byrne meant the US was losing 1-2 and that meant Fischer had to beat Gligoric to salvage a third place finish. He couldn't do it and lost and as a result Argentina finished a well deserved third instead.
Fischer and Botvinnik have pretty much slipped into obscurity as the chess world moves on and the chess understanding of today's top players has surpassed theirs, but at Varna in 1962, one of the most looked forward to games was the World Champion Botvinnik against the hopeful Fischer. At first it was rumored that Botvinnik would be given a rest day, but fortunately for the chess world he ended up playing. This game was the only one ever played between the two.
Both players collaborated on the postmortem and in Fischer's My 60 Memorable Games the notes were mostly Botvinnik's. In most cases neither player offered any moves to verify their differing opinions, so I decided to take a look at it with Stockfish 8 and Komodo 10. On the whole, the engines agreed with Fischer's opinions more than Botvinnik's.
Fischer gained an advantageous position in the middlegame, but gradually let his advantage slip. The ending after move 51 was interesting because both players reached different conclusions; In My 60 Memorable Games Fischer claimed that he had a win if he had chosen to play 51...Kd4 instead of the move he actually played. Some years later Botvinnik handed the problem over to one of his students, the 13-year-old Gary Kasparov, who claimed that he had found a draw as did Lev Zaitsev in independent analysis. Analysis with Stockfish seems to support the claim that the position was only a draw.
Back then overnight adjournments were common and the Soviet team did a good job finding the draw. The game was adjourned at move 45 the Soviet team went to work. Botvinnik was in a difficult position, and probably objectively lost. Tahl recalled how he, Boleslavsky and Spassky worked on it for hours. Botvinnik, Geller, Keres and Furman analyzed all night in another room. When Tahl went to the room of Donald and Robert Byrne to offer a draw in his adjourned game with Donald, he found the Byrne brothers working on the Botvinnik-Fischer game as well. David Levy wrote that it was Geller, who in the early hours of the morning, found the implausible drawing idea beginning with 47.Rxh7! where White allows his opponent two connected passed pawns.
You can watch the conclusion of the game when it was agreed drawn on Youtube HERE.