I've had very good success using nitromethane 30% used for RC cars. Also Lacquer Thinner, but the nitro seems to work better. It sort of balls up the black, not actually dissolve it. I do this on polycarbonate so may work different on what ever plastic your tiles are made of. Sample on a spare tile or the back side of one if you don't have a spare.
Here's an Amazon link I've bought from. I see it's currently not available, but I paid $14 last year, September.
****EDIT****
OOPS!
I don't have enough posts to add a link.
Here's the listing title;
Byron Originals, INC. Race ProDriver 3000 Gen2, 30% Nitro, 9% Oil, 1-Quart, BYR3130219
Just do a search for RC nitro 30% and you'll probably find some. Good luck!