Somewhere on the Hyperbolic Plane, you encounter ten mathematicians. You know that some of them always tell the truth while the rest always lie. Each of them knows which of the others tell the truth and which do not.
Let $g$ be the number of mathematicians that tell the truth. Let $h$ be the sum of the indices of the mathematicians that tell the truth (e.g. if mathematicians 4 and 5 tell the truth, $h=4+5=9$).
You ask the mathematicians for the values of $g$ and $h$ and receive the following answers:
Mathematician 1: $\quad g = 6 \quad \lor \quad h = 12$
Mathematician 2: $\quad g = 3 \quad \lor \quad h = 12$
Mathematician 3: $\quad g = 4 \quad \lor \quad h = 11$
Mathematician 4: $\quad g = 4 \quad \lor \quad h = 25$
Mathematician 5: $\quad g = 2 \quad \lor \quad h = 17$
Mathematician 6: $\quad g = 3 \quad \lor \quad h = 24$
Mathematician 7: $\quad g = 5 \quad \lor \quad h = 23$
Mathematician 8: $\quad g = 5 \quad \lor \quad h = 11$
Mathematician 9: $\quad g = 6 \quad \lor \quad h = 23$
Mathematician 10: $\quad g = 5 \quad \lor \quad h = 16$
Q: Which of the mathematicians are telling the truth?
Note 1: $\lor$ is the boolean "or" operator, so only one of the statements needs to be correct in order for a mathematician to be telling the truth.
Note 2: I constructed this puzzle using a search program that enumerates all possibilities to ensure correctness. It would therefore be nice to get solutions that do not rely on computer programs themselves.