You're right, but a clarification. In Armen's "The Gamesters of Pentathlon," the arena isn't an arena but a large, chess-type board (it's called a board in the script) with12 alternating black and white squares arranged in a 3x4 grid. The players can only advance one square at a time and only if they can disable the person on the target square.
In the climatic fight, Kirk, McCoy, Spock, Sulu, and Uhura face off against Shana, Taky, Roork, Klong, and Lars. The two teams move alternately. Kirk calls the moves for the Enterprise team and Shana calls the moves for the Pentathlon team.
That sounds like . . . they did a good job changing it.