Mahasattva: What is the Japanese phrase for politely asking for a game, i.e. "Please will you teach me?"
Richard Hunter Oshiete itadakimasu ka?
RichardHunter has given a correct and polite translation of "Please will you teach me", but I get the feeling that you are referring to what is commonly stated at the start of the game, "Onegaishimasu". What is the exact situation that you are referring to (that is, when would you say the phrase you want translated)?
Onegaishimasu is usually said before starting the game, after you have already agreed to play. It means please, but it means more like please be kind to me. It's more equivalent to "Have a nice game." It is not a way of asking for a game. To ask for a game, I would choose a phrase depending on the level of politeness (age, rank etc.). Ordinarily, I might say "Ikkyoku uchimasho ka?" To a much stronger player or a pro, a more humble request is more appropriate: "Oshiete itadakimasu ka." To a familiar partner, I might say "Yarimasho".
Mahasattva: My knowledge of Japanese is very limited but I thought that "Oneigashimasu" was the polite way of saying "please" when asking for help, as opposed to "dozo" when inviting someone to do something. So "Oshiete itadakimasu ka" is I think the phrase I was looking for. Does anyone know the origin of the custom of inviting an opponent in this way? Also what is one suppose to answer? Is there a set phrase?
Sensei's has a page with Useful Phrases In Other Languages, "other" being other than English.