Mizuwari (水割り) is a way of drinking whiskey or shouchu. Ingredient-wise it's basically whisky, water and ice. The interesting part is that it calls for a precise number of stirs at certain stages of making the cocktail. This is corroborated by Nonjatta, The Whiskypedia, The Star Online and many others.
The number of stirs given by most sources is 13 and half after you pour whisky over the ice and three and a half times after you add water. Nonjatta in his blog post quotes Nikkei mizuwari and Suntory mizuwari but none of the links are alive anymore. The last quoted page is Japanese Wikipedia for 水割り which contains no references.
Extensive googling provided me with another possible source Japanese Cocktails by Yuri Kato (Amazon) which is mostly sold out or fetches incredible prices (100+ USD for a copy) so that's out.
Where does the recipe come from then? Is there any reason for the exact number of stirs or is it there for the ceremony itself?