I've never actually played a sudoku puzzle (I know, I live in a cave!).
I just saw the word written in kanji somewhere, which rekindled my interest.
Wikipedia states:
The puzzle was popularized in 1986 by the Japanese puzzle company Nikoli, under the name Sudoku, meaning single number.
Though to me, the compound 数独{すうどく} implies "number addiction", especially in the context of a game.
Answer
I don't believe that sudoku exists as an actual word. Literally it does mean "single number," as it is a combination of the characters for "number" and "single." However the mathematical term for singular (versus plural) is 単数{たんすう}.
According to the Wikipedia page for it, the name is a reduction of the phrase "数字は独身に限る," or basically "limited to one number." I don't know if there are any hidden puns about number puzzle enthusiasts being forever alone, but a subsequent publication of this type of puzzle in Japan shortened that title to 数独, and thus it was named.
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