Proof that such a Turing machine cannot be constructed…

Prove there can be no Turing machine $M^*$ that takes input $n$ and:

i. halts printing 1 if $M_n$ halts on input 1

ii. halts printing 0 if $M_n$ doesn't halt on input 1

Intuitively I can see why such a machine can't be constructed, because in case ii. if the $M_n$ doesn't halt on input 1, we cannot determine whether it will ever halt or loop forever. However, I struggle with the construction of a formal/rigorous proof. I know problems of this nature generally work along the lines of: assume a machine can be constructed and show how it leads to a solution to the halting problem.

Any help is appreciated.

• So we would have (for i.) $M^*$ halt printing 1 if $M^*$ halts on input 1, which I guess is self-looping, right? – user1038665 Feb 26 '14 at 20:10
• This hint is a bit oversimplified. $M^*$ halts on input $1$ iff $M_1$ halts on input $1$. This does not lead to a contradiction. – Erick Wong Feb 26 '14 at 20:29