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As much as I wish this was my original idea, it's not. There was a thread about this subject a few weeks ago, and we decided that Bill would win with his heat-vision.
Shadowfax would win on the hypothosis that there would ever be a serious fight between them, but if there were a fight for not enough reason I could imagine him holding back, not being the naturally violent type, and then Bill would nail him because of his cunning and slightness and speed. Go Bill! But really I don't think they could ever possibly fight at all, but the thread doesn't deal with that question. However, I've always been fond of both these characters so my response is basically the same as 33Peregrin's. But I think I have to cast my actual vote for Sceadufax. Or is that Scaedufax? Does anyone else remember the exact spelling of the original Old-English version of the name?
Go Bill!
Bill would win. When Bill would come to fight against Shadowfax, Shadowfax would start laughing hysterically until he dies, or until he has to be disqualified. Ingolmo
Shadowfax would let drop a monumental Shadowfaxian roadapple, the likes of which there are no whicher, the smell of which would paralyze every living thing for a square block around, and that would be that!