I was playing a game, and felt sure that I'd found a set of numbers with an ambiguous solution. That is, there were 10 unknowns which could be filled in in one of two different ways, both giving a correct solution.

I stared harder and found a flaw, so it turned out to be untrue.

But it got me thinking, is there something in the mathematics behind it that means every correct set of numbers is necessarily and proveably unambiguous.

Hmm, thinking about that, the idea of 'ambiguity' depends on having some known and some unknown squares - you could argue that every correct set is a rearrangement of every other set if they're all unknown, unless you have some known ones that 'fix' the correct solution to only one.

So I'll modify my question to - does the sudoku application, and all the printed versions of them, assure no ambiguity of the answer? Is there (and I'll regret asking this) a technique by which the random ones are generated such that the given initial set give only one answer?