The trick is to retrace in dry erase marker what was just written using the Sharpie and then erase it away. I’m not sure the science behind it all, but as long as you cover the permanent marker writings with the dry erase’s ink, it will be erasable. Here’s a video below I made proving it:
The only reason the slate appears dirty afterward is that the eraser tip on my marker was a bit dirty already! As you can see, the permanent marker won’t erase on it’s own, but combined with the dry erase it easily wicks away. Next time you can’t find a dry erase or it’s running low on ink, a Sharpie can do the trick temporarily.