This dried paste is a porous material and can be difficult to clean. This guide is about removing permanent marker from grout.
How do I get ink out of grout?
By leanneg93 from Wenatchee, WA
Try putting rubbing alcohol. Put it in a well-marked spray bottle & spray it into the grout. Let it soak a few seconds then spray again, then absorb with a paper towel. You can also try the stronger 90% rubbing alcohol & also try fingernail polish remover. Use the stronger one that's NOT made for "artificial nails". Also, they sell professional ink removers (for janitors, etc) on the internet. (they work wonders!) Once it's clean, be sure to seal your grout!
How do you clean permanent marker off of tile grout?
Juanita from Stockton, CA
You might try a little bleach on a Q-tip, do a small area as a test. If that doesn't work try a Magic Eraser. Good Luck.