right - I did this last year and had superb potatoes.
I've copied this - it says with tyres - but I did it in an old plastic kitchen bin with a few holes in the bottom
week or two before your planned potato planting date, set your seed potatoes somewhere where they will be exposed to some warmth (between 60 and 70 degrees F.) and lots of light. This will induce them to begin sprouting.
A day or two before planting, use a sharp, clean knife to slice the larger seed potatoes into "seeds". Each seed should be approximately 1 1/2-2inches square, and must contain at least 1 or 2 "eyes" or buds. Smaller potatoes may be planted whole. In the next day or so, your seed will form a thick callous over the cuts, which will help to prevent it from rotting once planted.
If you have no established garden plot, or if there just isn't enough available space within your garden, you can still grow a respectable crop of spuds, and do a little recycling at the same time. Potatoes thrive in the warm environment of a soil filled tire!
Four tires + Two pounds of seed potatoes + Good soil = 20-30 pounds of winter potatoes!
Pick a spot where you can stack your tires which is out of the way and preferably out of sight. Loosen the surface of the soil just enough to allow for drainage, and set your largest tire in place. Fill the inside of the tire casing loosely with good topsoil, and then set 3-4 potato seeds into the soil. (Use sticks or rocks to keep the casing rings spread open.) Add enough soil to the tire "hole" to bring it to the same level as the soil inside the tire.
When the new plants are eight inches tall, add another tire and soil to the stack, as in the first level. Repeat the process for your third, and if desired, fourth tires. As you add tires and soil to the stack, the 8" of the plant stalk is covered with soil. By doing this, the existing stalk essentially reverts to a root status and the plant is forced to grow upward to once again find the sunlight which it needs. (much like if you were to try to eliminate a dandelion by covering it with a scoop of soil) By raising the soil level this way (in 8" increments) the plant is able to continue growing without suffocation, and at the same time you are creating a 24-30" tap root from which many more lateral roots can develop. Each lateral root can then produce additional potatoes (at 3-4 levels rather than the normal single layer). When you water, be sure that the soil is thoroughly moistened all the way to the base of the pile.