Tephrocactus is an interesting genus. Related to Opuntia, Tephrocactus grows in similar segments that are loosely connected to one another. A thorough drenching can make the segments of Tephrocactus articulatus var. papyracanthus (the “paper-spine” cactus) pop apart. That’s what happened to ours this summer, despite being careful with watering.
From what I’ve read, it’s tough to grow this particular species large enough to flower in cultivation. On the plus side, you get plenty of segments to root—they root very easily—so you can fill a pot quickly or start multiple plants.