Is chess a sport?
Hmm...I don't know how to answer this question, really :) They classify chess as a sport, I know that the International Olympic Committee recognizes it as a sport. But there are those who do not consider it a sport and there are eternal debates about that topic. On the one hand, mental activity (which is the case with chess) can very often be much more difficult than physical activity. And again, on the other hand, this group that does not classify chess as a sport does so precisely for the reason that there is no physical activity present. I have to be honest and say that I still belong to this second group :)
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Also, I think it depends on the culture too. For example, sports in Bengali is called "খেলা", which can be used for board games and video games too. But sports in Chinese is "体育运动", which translates to "movement of the body".
I think because of the concentration and training regime required, chess is considered a sport. I think it is more similar to esports than what are commonly regarded as sports. The latter have a physical, fitness element that people who say a sport has to have a physical component use to discredit esports and chess as sports.
I think to have a strong mind, discipline and skills, you need to do some sort of training to be able to sustain some long periods of focused concentration... So there is a part of physical activity required in mind sports too.
There is a documentary, called Mind Games - The Experiment in which they take mind sports athletes (esports, Chess, Mahjong and memory games) and have them go through a physical training regime to see the effect on their skills. All of them see improvement, so I think both are linked.
Then again, I've played chess and video games since I was very young so I'm a little bias :)