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Not that I have a better answer, I believe the phrase "a third of a pizza" is just as confusing.

You are, of course, quite correct.

I should make a mental note to comment on HN while actually focusing on something else..... :-)

Third can be used both as a noun and an adjective.

Noun form: "A third of a pizza".

Adjective form: "A third pizza".

You can say "a third of a pizza" and not be confused though, so there's no difference in saying "a second of a pizza" if that were indeed an accepted use of the word.

Secondth?