Is there an algorithm to decide whether the nth Busy Beaver number is even or odd? Or is this problem r.e.-complete? Or might it have intermediate Turing degree?
(For readers with social lives: “Busy Beaver” is not what you think. As discussed in this Wikipedia article and this old essay of mine, it’s the maximum number of 1′s that an n-state, 2-symbol Turing machine could write on an initially blank tape before halting.)