Use hydrogen peroxide. Need a recipe? Mix together 1/2 cup of baking soda, 1/4 cup of hydrogen peroxide, and 1 tsp dish soap. Apply cleaning mixture onto the grout, wait...
You'll need to use an abrasive bit and word very slowly. Keep the drill bit wet, because it's going to get hot very quickly. Once you get through the top...
You can use a regular household drill, but you'll need to use an abrasive diamond drill bit. Use the drill on low speed, and keep the drill bit wet with...
Use an abrasive diamond drill bit, and work with your drill set to the lowest speed. Go slowly, and frequently dip the drill bit in a little bit of water...