I thought new potatoes could be grown from the "eyes" on a potato? You know how they sprout little roots when you forget about them?
I remember starting sweet potato vines from a potato with tooth picks stuck into it around the middle and stuck in a mayonnaise or pickle jar. We always did them in elementary school and would have them lined up along the classroom window ledge to see how long it would take to sprout and who's would get the biggest.
As hot as it's been, put a couple into a paper bag, stick in the garage and see how fast they sprout. I keep onions in a bin on a shelf in my garage, and I noticed the other day, that one of them had sprouted. Only bought them a couple of weeks ago. Has to be the heat.