Wikipedia and some other lists include quite a list of possible answers. Except that a number of them are NOT possible. In fact, most of these articles fail to make ANY reasonable attempt to weed out even the most impossible answers, which makes them very unhelpful
First, note that the expression was well-established in restaurants by the early or mid 1920s --along with other short # codes used by waiters and waitresses-- as shorthand for something that could not be served because it had run out. Apparently it was later extended to mean "refuse to serve" and even to kick out/bounce someone.
Note the date. The simple chronology rules OUT explanations such as the following:
a) explanations based on things that did not even EXIST at the first known use of the expression (by the mid 1920s) -- Chumbley's went up in 1927, the Empire State Building in 1931, the New York Liquor Authority in 1934 (Other supposed sources are as late as the 1950s!)
Limit on # of beds or portions, related to Great Depression era soup lines, etc. won't work either, since the Depression began in 1929.
b) explanations thats depend on the LIQUOR connection ('stop serving') which is a LATER derived use (no earlier than the 1930s)...
And too many depend on claims that no one offers ANY source for (such as a purported law never actually cited or a general reference to "A restaurant" or "a morgue"...)
c) connecting it with the Uniform Code of Military Justice section regarding AWOL soldiers.
Now it is true that UCMJ Sub Chapter X Article 86 treats this subject.
http://en.wikipedia.org/wiki/Uniform_Code_of_Military_Justice
Only problem -- this passed Congress May 5, 1950, signed into law by President Harry S. Truman, and became effective on 31 May 1951.
Other problems
-- various suggestions based on only 85 of something ever being prepared or available offer no evidence for this number (an unusual one at that)
-- I don't think the British Merchant Marine ship crews of the early 20th century were anything like that large)
That probably leaves us with things like
- rhyming for "nix" (meaning "nothing", that is, we're out)... "nix" was borrowed from German "nichts",and was around before 1920.
- the famou Delmonico restaurant's menu item #86 --rib eye steak-- was so popular that they often ran out.. so that "86" came to be used for "we're out of that"
- a Morse code abbreviation ? (the system began in 1859, but I have not found any list including "86" yet)
At this point, I'd have to save the rhyme with "nix" seems more likely than others... but more from the WEAKNESS of the other explanations!
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Extra piece of trivia -- this expression was the origin of the decision to make Maxwell Smart "Agent 86"