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This book states that "a player who is furiten, is not allowed to win on a discard." What does furiten mean, and how does it affect the game?

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Furiten is when you are unable to win on a tile due to certain circumstances. There are about 3 different circumstances.

One type of furiten is when you previously discarded a tile that you needed to complete your hand. Let's say if you go for a hand without honors, 1s or 9s(a hand called 'tanyao', worth 1 han), and you have a double wait of 3 or 5 circles to win. If you have previously thrown 3 circles, even if it was taken by someone when they did a chi/pong/kong, you are unable to take the tile if someone else discards it. You are, however able to take it from the wall yourself and win with it.

The second type is when you ignore a tile needed to complete your winning hand. Similiar to the previous situation, but this time you are waiting for a 1 or 4 circles to win. You have neither thrown the 1 or 4 circles previously. Let's say someone throws the 1. If 'tanyao' is your only yaku, you obviously are unable to win it. So you pass on it. However, this puts you in a state of temporary furiten. If within the same turn(before you get to take another tile), someone throws the 4, you are unable to claim the tile for your win. If someone does a pong/kong, you will remain in furiten until your turn comes again.

The third type is when you declare 'riichi', and ignore a winning tile. Since riichi is already a yaku, you are able to win even without any other yaku. Let's say you are waiting for a 1 or 4 circles(again). You have neither thrown either tile previously. Someone throws the 1. If you don't claim it, you will be declared furiten for the REST OF THE ROUND. Same goes if you draw a 1 and discard it. In other words, your hand will be permanently declared invalid til the next round.

source: i love mahjong~ :D

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Thanks for the thorough answer, and welcome to the site. – Don Kirkby Dec 21 '12 at 20:48
Nice answer! Could you clarify what "rest of the round" means? (Until someone wins the current stage and the wall is rebuilt, or until its your turn again after supposedly 3 turns by the other players?) – mafutrct♦ Dec 22 '12 at 18:31
thanks for the welcome. and the rest of the round means til someone wins or the tiles run out. after the wall is rebuilt, and you take your tiles, that is when a new round starts, even if the round still remains the same due to the dealer winning or waiting for a last tile(tenpai). – Yuri Dec 23 '12 at 11:19
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According to the mahjong wiki:

Furiten in riichi mahjong is a state in which a player has discarded a tile that would have otherwise completed his hand. A player who is furiten cannot win on an opponent's discard, but can only win by self-drawing the needed tile. This rule is the crux of riichi mahjong. This is why all discards and melded sets are organized to show ownership the tiles discarded.

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It's not just in Riichi... – aramis Jan 21 '12 at 9:37

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