Growing a Three Sisters Plot
First off, a quick review of the Three Sisters- this is the shorthand term for the farming practice of growing corn, beans (pole beans) and squash all together in the same plot. This was practiced by many of the Native tribes of the Americas.
So...secondly, how to do it? The principals are simple. Pole beans need a trellis to climb on- so they get planted with the corn. Ideally, a corn variety that still grows a vigorous stalk. Since beans harvest their own nitrogen from the atmosphere, they put very little demand on soil nitrogen, making them less competitive with nearby corn. Finally, squash is grown in the understory beneath the corn and beans- ideally a heavily vining type, as there are modern squash varieties bred to "bush culture" where the leaves and stalks are far less numerous. You want, ideally, an old-fashioned vining type that will naturally fill out all the available space. Big squash leaves harvest sunlight before it hits the ground, reducing evaporation and making a moist environment that lets all the roots grow more consistently- while also shading out weeds. They both inhibit or stop weed seed germination, or directly crowd weeds that do manage to germinate.
I plant the corn in "hills"- which is an old fashioned way of saying on a grid system, with clusters of plants. Depending on soil type, you won't actually be making real hills. In overly draining soil, you will actually want to be planting the corn in minor depressions to collect water. "Hills" is just to say that you will be planting 5-8 corn seed together in around a square foot or so of ground.
The bean seeds are sown 1 to 3 weeks after your climate's appropriate corn date, and should be sown within a foot of a corn hill. You can sow anywhere from one to three bean seeds per corn hill (though I'd say 2 plants is a more reasonable target), and you can sow as many hills as you want, depends on how much you love beans! They are the most laborious to pick and shell by hand, so don't overshoot your bean ambitions.
Roughly equidistant from each corn hill, you should plant your squash hills. These may be 1-3 seeds per hill, though you ultimately want only 1 or 2 plants per hill. If you get proper vining types, they will quickly cover most bare soil in all directions by late July.
There is a common misconception on the internet that somehow the Natives just planted these 3 species and everything worked out perfectly, with little to no weeding. It's just not true. I've carefully studied the actual old practices of the Haudenosaunee of New York, and they were careful, assiduous weeders. By late July your field access is very difficult, due to corn height and squash foliage, but up til then, you should be frequently weeding- in most climates, this is most efficient with an old fashioned hoe meant to heap dirt up on the cornstalks. Scraping the soil kills the weeds there, and when heaped on the corn, it should bury and kill the majority of the weeds there. Several imperfect passes are better than 1 perfect pass. There are always more weed seeds that will germinate- better to strike fast, and often, and move quick, and come back another day to whack the survivors. The old timers knew the way and didn't mess around with the newest Jo&nny's gadget where steel has been bent in some newfangled shape and sold to you for another $40-$60. The Natives did it with hoes made of deer, elk, or moose shoulder blades, and the old carbon steel hoes of our grandparent's time are the peak of that level of technology- you are moving soil and you mean business. Put away the scuffle hoes, those are great for your lettuce, not needed here.
Interestingly, even more complex polycultures existed, including the Aztecs adding, at various times, amaranth, chiles, and tomatoes. By understanding the interplay between species, you can come up with your own successful polyculture combinations. The principals are simple- if sunlight is landing on bare ground, weeds will show up. So fill out the space and pay attention to angles of sunlight, what species need full sun, what does well in partial shade, etc. This past year, I was worried about vining squash leaping out of my Three Sisters plot into the potato and garlic plots on either side, so I replaced the "squash" component of the outer rows with large chile plants- one in between each corn hill. They grew tall enough to stay above what squash leaves came their way, and were far enough from corn to get their own sunlight, as heat-loving crops. Being on the border, I could visit them frequently to stay up on picking the red chiles. Following those principals, you can intercrop all kinds of things together.
You don't quite magically get the full yields of all the crops involved. My quarter acre of Three Sisters does not result in a quarter acre's worth of corn, quarter of squash, etc. Studies show that you get about 6/10ths the yield of corn and squash, which means that if you had 4 parcels to plant, and one person planted separate plots of each species (2 and 2), and another planted all four parcels to a Three Sisters style, the Three Sisters farmer would harvest 2.4 units of corn or squash for every 2 units of corn or squash of the monocrop farmer. On four plots, that means the monocropper gets 2 units and 2 units, the polyculture Three Sisters gets 2.4 and 2.4. That's not counting the beans, whose yield is a bit more difficult to express.
If you're wondering...why in the hell doesn't everybody switch to this? It's simple. Three Sisters is more labor intensive. The polyculture is not easily done by machines- it's mostly hand sown, hand weeded, hand harvested. Perfect for the gardener, homesteader, or farmer, whose primary consideration is space, not time.