Here by «algorithm» I mean a Turing machine with finite language and memory.

Is it possible to obtain by purely existential (i.e. non-constructive) means the existence of an algorithm performing some required task, while beeing unable to explicit it (i.e. write it down or describe it constructively) ?

I hope the question is neither too vague nor too ill-posed, as I'm not so acquainted with these concepts! Thank you for any insight on the matter, feel free to retag or plainly close this naïve question if necessary.