The best way to clean old wooden floors in an old house?

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We rent an old house that has wooden floors, not hardwood floors, just old wooden floors with some old paint in places. I clean them every day but they are still dirty. I am at my wits end because my grandaughter doesn't like to wear shoes and her socks and feet get dirty even though I mop the floors. I have tried mopping the floor with many different kinds of cleaner and event tried using vinegar. I have tried bleach, pine cleaner, cleaner made for wood. This is a rented house and we hope to move hopefully sometime in the near future but for now I need clean floors. I have some throw rugs scattered about. But, I would like to remove some of them but don't won't people to think that my floors are dirty. Need some hints.