I am a college student and at a recent function they were giving out small cacti in pots. I took a few home and soon discovered they A) had no root system but were just placed on the soil and B) were in regular gardening soil, not cactus soil - so they had little drainage. How can I best get them rooted and healthy once again? One has already died (it shriveled up, it looks as if it got too dry), and I don't know how to save the poor little things. A friend recommended a partially dampened sponge to allow them to root, so right now I have them on a partially-dampened paper towel.
You may have been given 'pups' or off shoots of a larger cactus plant.
This would be why they do not have any root system at this time.
The link below will help you with starting these new cactus.