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Toncha - cha here stands for seating (as in, the player in giving seating). Each player, from dealer to last, is: east (ton), south (nan), west (shaa), north (pei). English speaking circles rarely actually use the terms toncha - nancha - shaacha - peicha, generally just using east/south/west/north player.
Haipai - starting hand. The hand you draw in the beggining of the round.
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Yakuman - higher limit hand. Mangan > Haneman > Baiman > Sanbaiman > Yakuman. Those terms generally are left untranslated, so people just learn those. They're pretty common, tho, so it's not hard to learn them as they are.
Those limit are applied as a way to not let the hands score too high given the exponential nature of the scoring system of the game. Yakuman is the highest one, and it's really rare to score one of those.
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Madoka's confusion is explained by the twist of this chapter. She's basically waiting for the MC draw his first tile. He doesn't draw it in the end, so he just stays all of this round with one less tile.
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Pon - one of the types of calls you make to use the tiles of your opponents to complete and speed up your own hand. With pon, you can make triplets (a set of 3 identical tiles).
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Chuu is actually a mistranslation. The 中 tile name is actually chun.
Hadaka Tanki - if a player make calls to complete all of their hand, then the last tile in his hand will be his pair wait. Hadaka just mean "naked", cause everyone can see that this player has a tanki machi (pair wait). It's not so common for people to go for hadaka tanki cause it's easy to defend from it, and having only one tile in the hand makes it's to hard to defend from attack from others.
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"The first oya tsumo" - Oya is generally called "dealer" in the english speaking community. The dealer is the first player to draw and discard, and it's not so uncommon, mainly with people not accostumated with playing with real tiles, to forget to make the first draw, what makes them unable to complete their hands. As MC didn't make the first draw, he has one less tile in his hand, so, after all the pon calls, his hand becomes empty. - Tsumo is a generic term for the act of drawing a tile, so it's just translated to "draw". As there exists a yaku also called "tsumo", calling generic draws "tsumo" generally confuses people in the english speaking community.
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Agari Houki - Generally called "dead hand" in the community. Being unable to complete a hand because of a fault.
Tsumogiru - Tsumo = draw. Kiru = discard. Tsumogiri is the act of discarding the tile you just draw (instead of one from your hand).
Red woo pin - Red 5-pin. 5 is generally romanized as "uu", but the english speaking community doesn't generally use the japanese names for the tiles. Red dora (Aka Dora) are a special version of the 5's of each suit that give bonus points for having them on your hand. As Kamiya completes her hand with this tile, she wins the bonus. Dora tiles generally are dangerous to discard because of this, but here the MC has no choice.