I commented it below too, but then I read your comment. I've found Hive is very enjoyable for a chess player (myself) since the strategy is different as the board is the pieces. If you want more level of skill, Terra Mystica was also very strategic balancing territory and resource control.
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Strategy games which are simple to explain, but allow skilled players to compete without getting bored.
Go, Scrabble, chess... What modern games are like this?
Or, silly games where losing doesn't matter, it's fun anyway. My current favourites are Space Alert and Galaxy Trucker.
I commented it below too, but then I read your comment. I've found Hive is very enjoyable for a chess player (myself) since the strategy is different as the board is the pieces. If you want more level of skill, Terra Mystica was also very strategic balancing territory and resource control.