Scuff marks on the floor are inevitable -- even for your grandmother, whose house always seemed in pristine order. You can't expect your beautiful linoleum, considering all of the feet and paws that walk on it and all of the furniture you drag over it, to stay perfect forever.
So what was your grandmother's secret to keeping up the appearance of perfection? It was quite possibly the trick of using old-fashioned toothpaste (not the gel kind). Scuff marks on linoleum can often be removed if you use an old toothbrush to work the toothpaste into the mark, wipe it away with a damp cloth, then wash the floor as usual.