1. However brilliant she is, she won't be able to solve this math problem.
2. However brilliant she may be, she won't be able to solve this math problem.
3. No mater how brilliant she is, she won't be able to solve this math problem.
4. No matter how brilliant she may be, she won't be able to solve this math problem.
(What expressions are grammatical? Are all grammatical and the same? Then which ones are commonly used? Thank you for your help.)
All sentences are fine and all mean about the same thing. The only correction needed is the spelling of "matter" in #3.
All of the given expressions are grammatically correct, but they have slightly different meanings.
1. "However brilliant she is, she won't be able to solve this math problem."
This sentence implies that the person is indeed brilliant, but it suggests that their brilliance may not be enough to solve the math problem.
2. "However brilliant she may be, she won't be able to solve this math problem."
This sentence suggests that the person's brilliance is uncertain or hypothetical. It acknowledges the possibility of their brilliance but still concludes that they won't be able to solve the math problem.
3. "No matter how brilliant she is, she won't be able to solve this math problem."
This sentence implies that the person is absolutely brilliant, and regardless of their brilliance, they won't be able to solve the math problem. It emphasizes that their brilliance doesn't affect the outcome.
4. "No matter how brilliant she may be, she won't be able to solve this math problem."
This sentence also suggests uncertainty or hypothetical brilliance. It conveys the same meaning as sentence 3, but with the added notion that the person's brilliance is not confirmed or known for certain.
In terms of usage, all of these expressions are grammatically correct, but sentences 3 and 4 are more commonly used in everyday conversation. They convey a sense of certainty and emphasize that the person's level of brilliance doesn't matter in relation to the math problem.