Thomson was born in Glasgow, Scotland, the youngest of six children. He arrived in the United States two years later. His father, a cabinet maker, had moved to New York City shortly before Bobby's birth and sent for his family in 1925. Thomson grew up on Staten Island in New York City and signed with the New York Giants for a $100 bonus right out of Curtis High School in 1942.

Longstanding rumors that the Giants engaged in systematic sign stealing during the second half of the 1951 season were confirmed in 2001. Several players told the Wall Street Journal that beginning on July 20, the team used a telescope, manned by coach Herman Franks in the Giants clubhouse behind center field, to steal the finger signals of opposing catchers. Stolen signs were relayed to the Giants dugout via a buzzer wire. Joshua Prager, the author of the Journal article, outlined the evidence in greater detail in a 2008 book.  Although Thomson always insisted that he had no foreknowledge of Branca's pitch, Sal Yvars told Prager that he relayed Rube Walker's fastball sign to Thomson. Branca was privately skeptical of Thomson's denials but made no public comment at the time. Later he told The New York Times, "I didn't want to diminish a legendary moment in baseball. And even if Bobby knew what was coming, he had to hit it.... Knowing the pitch doesn't always help." Whether the telescope-and-buzzer system contributed significantly to the Giants' late-season 37-7 win streak remains a subject of debate. Prager notes in his book that sign stealing was not specifically forbidden by MLB rules at the time and, moral issues aside, "...has been a part of baseball since its inception". Sign stealing using optical or other mechanical aids was outlawed by MLB in 1961.  Thomson however, late in his life, rebuked his prior claims of not receiving the sign on the Shot Heard 'Round the World, and admitted that the sign was received.

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