I think that if it is the Mexican Old Man cactus, they are very susceptible to rot. That's why you don't see very many in collections. I've managed to keep mine alive for about 5 years (at least I think it's alive). I don't water it at all in winter and just once month in summer. I had a really nice one, but I gave it to my son's then girlfriend--it probably hit the garbage sooner than later. If I would have known they were going to break up, I never would have given it to her :-). Can't help you much, but do let it dry for quite a while before you try to root it. And a pic would be good since there are a few different cacti called old man.

Succulover, do not apologize to me. If I felt I needed to apologize for every time I had a wrong notion, I would have used up all the ink in cyberspace. The internet would not be possible. I have so many wrong IDs that RoRo's eyes are permanently stuck in a roll, as in she probably rolls her eyes when she reads my mad-stab attempts at identification. brewerbonsai, sometimes the larger plants do not root from cuttings. I have found that plants like Notocactus root much easier than Espostoas, Oreocereus and others. Old Mammillarias might not, but the same species somewhat younger do. If the cutting has gotten hard (time to stick a pin in it?), and no roots are evident, then re-rooting is unlikely. Sometimes rooting waits until spring, so if the plant is still soft, there is hope.

Usually rot will show up quite suddenly, all at once. I don't think a heat mat will be necessary- you need warm, not necessarily hot. I also don't think there is much need for a heat mat in general. Of course, it all depends on your climate. If you can, grow the cacti outside (once they are rooted) in full sun. To do this, you will need to acclimate them to the higher temperature and sun levels by leaving them outside for increasingly larger periods of time. In other words, slowly increasing the light levels. I would start only about 1 hour outside, progressing upward maybe an hour every 1-2 weeks, until you can keep them outside permanently (if your climate doesn't allow this, put the cacti in a south or westward facing window). Your problem with the rot was most likely overwatering. Although I am sure it didn't seem like you were watering much, with the soil that cacti usually come in (compost, topsoil, and usually a lot of peat), the top couple of inches can be bone dry, while in the middle it is extremely wet. That is what makes this kind of soil so undesirable. To test when to water, you could use a wooden skewer, and poke it into the soil as far as it will reach, and if it comes up moist, you will know not to water.

In my opinion, the second one is goner. However, you can save it in a way. Get a sharp, clean knife and cut the top off above the bad area. The interior should look damp and clean with no sign of the deteriorated area of the plant. If you still see a bad spot, cut again until you see all clean tissue. Set it aside out of the sun for at least a week so the cut edge can dry up. or callus. Get cactus and succulent mix from the nursery (you can get a better mix if you also get perlite or pumice to mix with it 50/50, but at the least get commercial cactus/succulent planting mix). When the bottom of the cactus is completely dry, plant it in the mix you got. Do not water until your cactus has started to grow roots, which could be weeks. Don't worry, it has a lot of moisture stored so it will survive. If you water, it will rot because it has no roots to take up water. As for pots, you could use that little pot it was in if it has holes in the bottom and if you do not double pot it as it was. If you buy a pot, get a small clay pot. Your plant does not want to be in a large pot. It died because of being double potted and the soil isn't fast draining enough and was holding too much water.. Also, it was probably not getting enough sun. It needs full sun, but transition it gradually to that much light. Others know more than I, but I would repot the euphorbia in a slightly larger clay pot. The little buckets are cute but are not helping your plants survive.