Although Clayton is one of the best chess players in the world, during his simultaneous exhibition, one of the people he is playing beats him after playing only two moves, a losing combination known as the "Fool's Mate" and notorious for being the fastest possible way for a player to be checkmated. This, presumably, is meant to demonstrate how much Columbo is unnerving him.
During his simultaneous exhibition, Clayton says that Frank Marshall once played 155 games simultaneously, losing only eight, and later having complete recall of all the moves from all the games except two of them. This actually happened in 1922.
The final combination of the chess game played between Dudek and Clayton in the restaurant and finished in Dudek's hotel room actually was played in game Wolthuis-Alexander, Maastricht 1946. Dudek demonstrated the line leading to mate; in the actual game, black resigned after the first move of combination, Qxb4. Also, Columbo, when reading from Dudek's notation, says that Black resigned on the 41st move. In the actual game from 1946, the sequence takes place earlier in the game - Black resigned on the 25th move.
Laurence Harvey, who played Emmett Clayton, had a high fever during filming. A heavy smoker and drinker, he passed away on November 27.1973 at age 45, the same year this episode was released.
Charles Clement, the episode's technical advisor was also Universal's long-time head of post production operations and a life-long chess enthusiast rated by the US Chess Federation. He provided the dialogue, board set-ups, and props for the chess sequences, and adapted all of the game play and positions from actual historical games of note. Clayton's simultaneous match was similar to an actual 1964 simultaneous exhibition by American chess grandmaster Bobby Fischer (U.S. Champion at the time), in which Clement played Fischer, ending in a draw. In 1972, challenger Fischer defeated defending World Champion Boris Spassky (USSR) to become World Chess Champion. Fischer and Spassky likely provided the inspiration for the episode's leading characters, Clayton and Dudek. At the climax of the simultaneous exhibition sequence, Clement's right hand moves black's Q-h4 #, with the challenger declaring, "I'm afraid that's check, and mate, sir."