Those games don't really improve any sort of skill, though, and neither does anyone expect them to. To teach kids this, you need a game where you as a player pretty much never stop improving, so that having spent more hours on the game actually means you'll beat anyone who has spent less.
Go might work.
There are schools that teach Go intensively from an early age, so that a 10-year-old student from one of those schools is already far better than a casual player like me will ever be, and it just keeps going up from there. People don't seem to get tired of it.
Every time I contemplate that, I wish all the talent thus spent, could be spent instead on schools providing similarly intensive teaching in something useful like science and engineering. What could be accomplished if you taught a few thousand smart kids to be dan-grade scientists by age 10 and kept going from there? I think it would be worth finding out.
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