Given: P && ^P (that means: P and not P)
Prove: Q (any unrelated assumption)
And you can do that, knowing nothing about Q. Here's how:
Q || ^Q (Tautology -- every proposition must be either true or false)
^Q => P (Tautology -- if P is true then anything implies P, by definition of logical "implies")
^P => ^^Q (Contrapositive of the previous -- if ^Q implies P then ^P implies ^^Q)
^P => Q (not not Q is the same as Q)
You know that ^P is true from the given: (P && ^P).
So, from only (P && ^P) you have proven the unrelated proposition Q.
It's easier if you do a proof by negation, of course. But we were explicitly not allowed to do that on this problem, since it'd make it trivial.