I would go with 1 plant per square, based on the size of the plant once it's grown.

I just read an old OG article on square foot, and potatoes were in there.
Use one potato per sq ft.
Dig down 12 inches, being sure to clear out the whole square.
Poke the bottom of the hole with a fork to break it up a touch, and put in a couple inches of compost.
Put yer tater in the center of the square.
Mix the dirt you dug out with more compost.
Only cover the tater by a couple inches- as it keeps growing up, you keep adding in more of your dirt/compost mix till the hole is full.
In essence, your spreading your tater up, sort of trellising with dirt, lol.
I haven't tried this though.. Haven't gotten into growing potatoes yet.

two years ago i did a yukon gold in a 5 gallon swc with a bubler in the middle. I used clay pellets from my hydroponics stuff. The top bucket sat on the bottom bucket's lid with the top bucket having drain hoes into the bottom bucket. The bubler tube ran all the way to the top of the top bucket. This was just a trial, but it did work. One tater was cut into three pieces with two eyes in each piece. They were placed in the bottom of the top bucket and covered with three inches of grow rocks (clay pellets). When the taters grew up about three inches, I would cover again until they were all the way to the top of the bucket.

As I said this was a test, but I did get a small mess of taters.

This year I am planning on two, 2x2 beds by themself. I am going with 2x6 boards, with one layer to start and then add 2x6s as they grow. i have vertical 2x2 stringers in the corners to screw the next layer of boars to as they taters go up. This will be about 5 or 6 boards tall. Planning on using russets in one and red pontiacs in the other.

In my 3x6 bed, I am going to put yukon golds in a square and then build up from there.