@someone_random , based on decades of home growing potatoes including conducting experiments and conducting research this is what I have found:
1) Each eye ideally should have 2.5 ounces of potato attached with it to supply the energy needed for growth and production.
Try planting an eye with just the potato peel sometime and you will see. It will sprout but never produce anything but very small new potatoes and very few of them. Also, try planting one of those marble sized new potatoes and see what you get...again very little and very small.
Cut pieces of seed potato should each contain an eye and weigh about 2.5 ounces.
2) When planted, each cut piece should be spaced about 12 inches from the next one. Closer spacing generally yields smaller and often fewer new potatoes.
3) The cut pieces of seed potato should be well dried and coated to protect against rot before planting. For that, I use sulphur. I was taught to use wood ashes for that coating but wood ashes are alkaline and potatoes prefer acidic ph.