Well, this is the parent. It was a gift from a student of mine 25+ years ago.

lgteacher, yours is an Opuntia (or Cylindropuntia). It should be ok. Just keep it moist. They're pretty quick growers and can be pretty tough little guys. Just don't let it dry out too much or for too long. kittikity, Can't really tell what yours are yet, but I would put them in some cactus soil as well. I don't usually use plain commercially sold "Cactus Mix" because if you look at lgteachers picture, you can see lots of organic matter, like small wood chips or twigs. Normally this is not good, but will help you keep your seedlings moist. Keep them moist. Not wet but moist. At this stage of life, they prefer high humidity. Many people grow their seedlings in plastic bags to keep them moist. I usually keep my larger seedlings (out of the baggie) in a large "tupperware" type container with the lid either partially or completely off, to maintain more humidity than just a pot sitting on a shelf. If they dry out too much and for too long, you will lose them :( I do have a friend that will put any small seedlings he receives or finds BACK into plastic bags. So,you could try that too. IF you happen to find algae growing on the top, put a fine layer of sand over it. Would love to hear your progress and see how they all develop (both of you)

Thank you for the comments on the pot. It is one of mine I did a few years back. I have been thinking of going back into ceramics as there are not a lot of what I call nice pots especially for cactus.

That's got one of the finest common names in Cactusland, and I'm not a fan of common names, but 'Dogtail Cactus' is one I refuse to eschew.I think it's Deamia (Selenicereus) testudo, and it's a tropical cactus which has one of the largest flowers in the Family. If I'm wrong about the specific name, it's a Selenicereus for sure. I wouldn't remove the pup now - wait a year or two, as it may help it flower sooner and it's not the right time of year to be removing it to grow. I'd repot it in porous soil - this in one of the plants which it generally is OK to use commercial cactus mix with, as it's more accommodating of peat-based soils, being tropical. Yes, repot it and wait a few days and give it a little water. This is the slow time of the year, so water it about once a month and keep it warmish and fairly well-lit (as it's junglish in origin, it's a plant that doesn't need direct light, just lots (preferably) of indirect light). When it's warmer it prefers outside, and you can generally water it every week or so when it's spring/summer/early fall. Personally, I'd leave the pup on next spring and cut that long stem in sections and root them - the plant would benefit from removing that irregular growth, and you'd wind up with lots of this great plant.