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In The Gamesters Of Triskelion, how many disembodied brains exist on the planet? I seem to recall after the landing party gets their colours, Galt said something like "red, you are on a good team". However, at the end we are only shown 3 disembodied brains. It would be a very strange way of talking to say "you are on a good team" if there are only 3 teams. However, I may have been mistaking as I couldn't find anyone saying "they have a good colour" anywhere in the transcript.

In any case how many brains are there, do only 3 brains make up the entire race?

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    Just because there are only three gamesters doesn't mean they control only three teams. Each gamester may have tens or hundreds of teams under their control, and part of their skill consists in organizing which of their teams to play against which of the other gamester's teams. – DJClayworth Nov 13 '14 at 22:06
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A triskelion is a three-pronged/three-part spiral, and has existed as a concept since at least 3200 BCE. For example, here is an ancient Greek jug featuring a triskelion, and a Celtic cup with a simpler version, from that Wikipedia page:

Greek triskelionCeltic cup

And they themselves use a three-pronged spiral for the Triskelion planet:

Triskelion of Triskelion

Since there is no indication that there are any other Providers, and the name of their planet is symbolic of "3", it would appear safe to assume that there are indeed only 3 Providers on the planet. However, it is unknown if others of their species exist on other planets.

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