Because of the list of correct answers for a finite subset of the Halting Problem is finite, the Turing machine you mention does exist, but we not only can't find it, we can't know when we've found it!
Where the Math Overflow post mentions that "experts could compute the particular value of n", there has recently been such a bound published in a thesis, such that we can be confident that we can never construct or recognize the solutions beyond that point (using a particular axiomatization of mathematics).
You referred to the idea that "you can have a Turing machine that solves this", and we can agree on that with the caveat that, above certain problem instance sizes, you can't know or verify in any way that you have such a Turing machine!
http://mathoverflow.net/a/153106
Because of the list of correct answers for a finite subset of the Halting Problem is finite, the Turing machine you mention does exist, but we not only can't find it, we can't know when we've found it!
Where the Math Overflow post mentions that "experts could compute the particular value of n", there has recently been such a bound published in a thesis, such that we can be confident that we can never construct or recognize the solutions beyond that point (using a particular axiomatization of mathematics).
You referred to the idea that "you can have a Turing machine that solves this", and we can agree on that with the caveat that, above certain problem instance sizes, you can't know or verify in any way that you have such a Turing machine!