The match game in the movie is named as "Nim". In order to win the game there is only one tactic including a system called "Nim Sum". If both players use this tactic perfectly, then the owner of the first move will eventually lose. Due to the same reason, it is always disadvantageous to start the game first.
When the Man X tells to the Female A that she is at his grasp, but still unreachable, he shows her a sculpture of Tantalus who tries to reach a grape. Tantalus was a a Greek mythological figure punished to suffer eternally in Tartarus by standing in a pool of water beneath a fruit tree with low branches, with the fruit ever eluding his grasp, and the water always receding before he could take a drink.
"Marienbad" is the German name of the spa town Mariánské Lázne, Czech Republic.
The characters in the hotel attend a performance of a Henrik Ibsen play, "Rosmer." It appears to be some variation of "Rosmersholm." Director Alain Resnais discovered Delphine Seyrig in an American stage production of Ibsen's "An Enemy of the People."