"Hold it, Jim" ~ The scene where Mitchell hops into the shutting turbolift doors with Spock and Kirk near the beginning of WNMHGB.
You deduced the correct scene, @RubySlipppers. You have the game.
My next clue was to point out that it was done in one, continuous take. The "corridor" outside the turbolift doors was constructed of "wild walls" that were set up on the bridge and removed during the turbolift ride to reveal the bridge when the doors opened. One of my favorite scenes:
MITCHELL: Hold it, Jim.
KIRK: Getting into shape?
MITCHELL: Yeah, well I figured you weren't on the Bridge. Kelso's voice sounded a little nervous. Well, uh, you finish the game?
SPOCK: He played most illogically. His next move should have been the rook.
Clue 1: Spock fails to understand the lesson of the game.
Spock's comments indicate that he refuses to believe that logical play can be beaten by Kirk's illogical tactics.
Clue 2: Final outcome does not involve a Queen.
It involved the Rook. The "Queen" chess passcode was used in the later episode,
Whom Gods Destroy.
Clue 3: Most of this scene happens on the move, but moving is relative to the ship.
First, Mitchell jumps on the turbolift which then moves within the ship to the bridge.
Clue 4: Running is good exercise.
Mitchell needed to run to catch the turbolift after which Kirk jokes with him about "getting into shape".
Clue 5: Smallest "room" on the ship.
The turbolift has walls and a door which technically makes it a room. It's only about 7 feet in diameter.
Clue 6: Mock concern over another crewman's stress level.
Mitchell's comment about Kelso sounding nervous is meant as light humor.
Clue 7: The scene features three though no blue or red shirts, but eventually, all three colors will be represented.
Of the three actors, Kirk wears command gold, Spock is also wearing command gold but for the rest of the series he wears blue, and Mitchell wears the "salmon" color which evolves into the red shirt for the rest of the series.
Clue 8: Non-usual ceiling lighting effect in this area during this scene.
I think this is the only episode (or one of very few) with the spotted grate lighting from the ceiling of the turbolift.
Clue 9: Close friends, only, for one of the occupants.
Mitchell is Kirk's closest friend at this point, and Spock will become Kirk's closest friend during the series.