Ari-ari
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Ari-ari 「アリ アリ」 refers to a ruleset which allows kuitan (open tanyao) and allows atozuke (winning without confirmed yaku). It is generally the most commonly accepted ruleset pertaining to Japanese mahjong.
Conversely, nashi-nashi forbids kuitan and atozuke. Nashi-ari would forbid kuitan but allow atozuke, and vice versa for ari-nashi.
Game settings
In addition to kuitan - Ari and atozuke - Ari, the following settings are often used in "Ari-Ari" rulesets: