The general idea here is that a potato plant will throw roots (potatoes) when the stalk is covered. The stalk will continue to grow as well. So if you can keep that stalk growing up, it’ll keep throwing new potato roots.
You need a stack of tires, five or so should do the trick, and some seed potatoes.
Cut the side wall out of one side of each tire. This becomes the top of the tire. Then add some drainage holes in the bottom side. You don’t want standing water in the tires.
Put the first tire on the ground, cut-side up. Fill it most of the way full of compost or soil, and plant three or four potatoes in it. You’ll want to cut each potato into pieces with two eyes at least.
Cover the seed potatoes lightly with soil, and then add straw on top. Water heavily.
Once the potato plants are about two inches over the top of the straw, add another tire, and fill it with straw. Repeat until you run out of tires or decide to harvest your taters.
You’ll end up with about 25-30 pounds of potatoes per stack.
You can do something similar by making a three sided box out of wood. Anything will work here, but pallets work great!
Instead of using tires, you just continue to add straw and a retaining board across the fourth side every time you need to add more height to your stack. Eventually you end up with a box full of straw and potatoes.
Enjoy your taters!