If the only thing the interviewer has to judge your skill on is what you wrote on the board, that's what will be used. When I'm interviewing, I'm trying to gauge how they approach a problem, how well they can reason through a solution, whether they see the assumptions they're making and the impact those have on the solution, what the limitations are of the algorithm they came up with, etc. It's really hard to see most of this if the candidate doesn't talk at all.