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The Fourth Doctor-K9 chess match referred to a game of chess the pair had together. In this instance it was K9 Mark I, and Leela moved K9's pieces for him.
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The Fourth Doctor and K9 frequently engaged in chess games together. In the middle of the game, the Doctor captured one of K9's bishops with one of his. The Doctor went on to tell K9 that one-dimensional chess exposes the limitations of the mechanical mind, before K9 put him in check. K9 computed he could achieve checkmate in six moves. In response, the Doctor exclaimed "Rubbish!" before quickly returning to the game.
The game was interrupted after the Doctor moved his king when Leela alerted the Doctor's attention to the fact that the time column had stopped moving. Fearing they could have fallen through a time spiral, the Doctor materialised the TARDIS to take a reading. Discovering they were on Pluto, the Doctor and Leela left to explore leaving K9 in the TARDIS. He later joined them.
Once the trio completed their adventure, K9 immediately returned to the game, moving his king, and reminding the Doctor of his calculation for checkmate in six moves. The Doctor then jolted the TARDIS, which sent the chess pieces flying across it. Attempting to contain his laughter, the Doctor apologised to K9, telling him they could finish the game after he reset the coordinates. (TV: The Sun Makers)
K9 Mark II later engaged the Doctor in another game, in which the Doctor moved K9's pieces for him. The game similarly ended in a presumptive victory for K9. (TV: The Androids of Tara)