Here is where the idea first arose:
Nick: They could be playing a board game! [Earlier in this session, we had mused over what Galadriel and Lúthien were doing in another scene, and Nick had somewhat jokingly suggested they be playing a board game, such as backgammon.]
Marie: Okay, fine. Do elves have chess?
Nick: A chess-like game.
Marie: Ok, if it were anime, it would be shogi, so do elves have...
Nick: There's a game they play in Vikings, there's a particular Norse game that you see them playing.
Marie: Skittles?
Nick: Might be.
Marie: I've never seen Vikings.
Nick: It is pawns on a board....
Marie: I think it's called Skittles. Wait, no it's not. What is it called? Because Skittles is more like bowling, maybe? No.
Marie: Nine Men's Morris?
Nick: Tafl
Marie: Tafl
Nick: Yep, that's it, that's the one. I've seen this. Okay.
Marie: Yeah, yeah, that's Norse.
Nick: Yeah, and Celtic, also. So something like that works, yeah, sure.
Marie: Alright, so, they're playing an elvish game...
Nick: But I kinda like the idea of it being pawns on a board. I don't care what all it looks like.
Marie: Yeah. It could be 3D chess from Star Trek!
Nick: Some production designer can figure out what that looks like.
Marie: So they're playing a game, with figures on a board, and...is it Mablung or Beleg? Does it matter to you?
Nick: I feel like Mablung is the guy.
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.{actual plot of the episode}
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Nick: So the way you do this, with board game scenes like this, is you have the character who is explaining strategy, and how sometimes you have to make sure you're in the correct position, and *boop* checkmate, you see? Or, you can turn it on its head, and have the person who is advocating for a more direct kind of approach win the game despite...because they did something that the 'chessmaster' character never considered was an option.
Marie: Yeah, yeah. So there's...
Nick: The question is, how tropey do we want to be?
Marie: Well, since the game is whatever we make it, the rules are whatever we make it, I don't think we need to go too chessmaster-y because it's not going to be a chess game. And the difference between Mablung and Thingol...they're not opponents. Like a lot of times, the reason you do a tense game situation like that is to have two people who would not otherwise be getting along duking it out over this chessboard.
Nick: Or a teach your student moment, like a teacher/student dynamic.
Marie: Right, right.
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.{Criminal Minds tangent}
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Nick: So the point is, there's this older guy and this younger guy, and he's constantly teaching him life lessons through the play of this chess game.
Marie: The dynamic between Thingol and Mablung should be that Mablung is very reliable, constant companion of Thingol's ever since the journey from Cuivienen, and that he kinda relies on him a bit, and that Mablung has been someone he can hand things off to. So the whole dwarf situation with Norn, Mablung kinda...did that.
Nick: handled it.
Marie: Yeah. Exactly. So Thingol's used to just being like, 'Okay, Mablung!' and then not worrying about it. Same with Beleg and same with Celeborn - Thingol delegates to these people. So that's the dynamic. So a friendly game of whatever is honestly just them enjoying each other's company and spending time together, I think.
Nick: Okay.
An example of a tafl board:
Nine Men's Morris:
And while I did not suggest it at the time, there is always the stone-counting game Mancala (if we want inspiration for the rules of our totally-not-actually-tafl elvish game):
Terrace:
(Unlike 3-D chess, this one has reasonable rules and is a game you can actually play; it's from Star Trek: The Next Generation)