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Bleach on floor tiles wouldn't be my choice. I'd use a bottled cleaner expressly made for floor cleaning that is stronger than Swiffer solution. There are alot of them available and I honestly don't have a real preference as they all seem to clean well. If one of those doesn't work I'd use something like Pine-Sol diluted according to the package instructions. In my experience, Swiffer Wet is great for general floor maintenance if used frequently but it's not good for cleaning really dirty areas (as you say.) If you do decide to use a bleach solution I'd test it in an inconspicuous area first and let it dry before doing the entire floor. (And whatever you use, I'd mop the entire floor with it while concentrating your scrubbing on the dirtier areas, of course. Otherwise, the floor is likely to look streaky.) It's also *possible* the tiles are wearing out in areas with a lot of foot traffic. If that's the case, part or all of the black you see isn't dirt at all but is due to the surface layer of the tiles wearing off.
It might help cut down on the foot traffic grime to put mats *inside* the entrances from outdoors. But daily living is dirty! Good luck!
What kind of tiles are they ??? Peel n Stick, or ceramic. Aloha!
Therese, I'd wash with dish soap and water. Let it dry, and try mop n glow, for a finish. Don't use a ton of mop n glow, you just want a shine, not a buiild-up. Aloha!