This episode is based on the fact that Martin doesn't play chess very much, he claims "Daphne showed me once", however in Can't Buy Me Love (1994), Martin is solving a chess problem and has a corn-nut as a bishop, since Eddie ate the real one.
When Frasier sets off the smoke alarm to purposely wake Martin, Daphne emerges from a bedroom across and down the hall from Martin's room. However, it has been well established previously that Daphne's bedroom is near the kitchen on the opposite side of the apartment from Martin and Frasier's rooms.
After Frasier loses yet another game to Martin, Daphne tells him that when one of her brothers would lose a soccer match, she'd cheer him up by making a sock puppet and having it sing to him. A true Brit would never use the word "soccer"; the correct term in England is football. Although in-world, this may simply be an example of code-switching.
When Frasier (Kelsey Grammer) and Martin (John Mahoney) prepare to play their final game, Frasier asks who should go first, and Martin replies that the person who lost the last game is supposed to start. However, according to the official rules of chess, the player with the white pieces always goes first.