Fold the sandpaper into quarters so there is a square edge to it. Use the square edge to sand inside the first gouge. Sand it down to bare wood, removing the shine and the stain.
Repeat for each gouge and indentation. Blow away the dust as you sand.
Use your thumb to press wood putty into the first sanded-out dent, packing it tightly. Use enough putty that it mounds up slightly from the surface.
Use an old credit card to flatten out the surface of the putty, pushing the card forward through the putty to level it out with the surrounding floor.
Let the putty set for 12 hours. Sand it by hand with your sandpaper to get it flat and dull. Wipe up the dust.
Brush a little polyurethane floor gloss over each of the areas where you used the putty. View the areas from an angle as you apply the gloss, so you can see where the dull areas are and cover them. Brush it very lightly and smoothly.
Let the gloss dry for eight hours. Buff it lightly with sandpaper. Add a second layer of gloss. Let it dry for 12 hours.