Almost all new tires come with white brand lettering on one side. Most of these tires have a blue paint over the white lettering to protect it from dirt and grease while the tires being installed on a car for the first time. The blue paint looks out of place, but it comes off with a little work and a few materials.
Apply the car wash soap directly to the blue paint. Don't dilute it with water in a bucket. Apply it full-strength, and rub it over all of the paint. Dish soap can also be used for the tires, but it's not for use on your car. Dish soap strips the wax off of painted surfaces, causing them to dry out.
Use the brush to scrub the blue paint. Since the tires won't get scratched, you can scrub forcefully to remove the paint, but if you scrub too hard in one area, the white brand lettering will also start to come off.
Rinse the tire often with hot water to check your progress. Scrub with the brush until all of the blue paint has been removed.
To get the white lettering even brighter after removing the blue paint, use a product such as Westley's Bleach Wite, which is made specifically for white paint on tires.