It is interesting to note that potatoes are the subject of a great many stories and cultural methods. I suppose many if not all of the methods will produce some spuds. Perhaps some work better than others?
I can only tell you how I do it. Plant in rows spaced 32 inches and plants about one per foot in the row. Just a guess here you see, as I don't actually get out the measuring tape. Plant the seed tuber 4 inches deep. When the plants show the first blooms, hill them about 3 to 4 inches (one time only). The tubers will be developing when the plants bloom and hilling keeps them from pushing up in the sunshine. If they see light, they turn green and get bitter. This is the only reason for hilling. A good heavy mulch will do the same thing. When the plants fall down and start to turn brown, it is time to dig them. OK, that is my story. I don't go for those methods that say to cover the leaves. One must remember it is the leaves that manufacture the food that is stored in the tubers. Don't cover up the food factory.
For your 4x4 bed may I suggest six plants. Two rows of three in each row. Fertilize them well. Water well, and mulch them deeply. Enjoy!
This is what you plant:
This is what you get:
As you can see the tubers formed at the depth of the set that was planted.
Gardening at 5000 feet elevation, zone 4/5 Northern Utah, Frost free from May 25 to September 8 +/-