Misato (OFF):“"Let's go for the next one. Janken, pon, pon, pon, pon, pon!"”
Again and again, they both tie (aiko), until finally Shinji makes a choki (scissors) and Misato wins.
Misato:“"Ahhh, that's too bad, Shinji-kun."”
Reichu: This is janken, the Japanese incarnation of "rock, paper, scissors". The only thing that really differs is the terminology: gu = "rock", choki = "scissors", pa = "paper", and aiko = "a tie".
Hexon.Arq: "Pon" is simply the Japanese equivalent of "pop", so I doubt it has any significance beyond that of our "kai, bai, bo".
Sharp-kun: Janken seems to come up a lot in anime. I've lost count of the number of times I've seen it in situations where I wouldn't expect it. For example, the newest series of Galaxy Angel has "The 13 Special Moves of Janken" between episodes.
Hexon.Arq: Well, "rock, paper, scissors" seems to come up a lot on this side of the planet as well. We're probably just so used to it that we don't really notice anymore. Its Japanese cousin's prevalence may simply equal that of the western Coin Toss.
