I just had a NO FREAKING WAY moment. I was outside milling around with the dog who was enjoying doing the same and decided just to end my experiment and pull the potato plants. The leaves were starting to turn a little yellow, and the stalks were starting to tip over. When I pulled the plants they had potatoes!
I made potatoes. And a lot of them. I buried a few cut up near rotten potatoes on a whim in a non-garden cultivated place in the yard, pushed some of the mulch around them over the peeping potatoes once and they turned into a few dozen edible ones. How does that happen? Nature is amazing.
Thank you all for your suggestions.