Buying and saving supermarket potatoes to plant in your garden used to work, that was before they began treating them with growth inhibitors which prevent sprouting and extend marketability of potatoes. The growth inhibitor stops the formation of 'eyes' or sprouts on potatoes and makes them nonviable for planting - they won't produce new plants.
In order to acquire seed potato you either have to purchase certified seed potato or save potatoes for replanting from the previous season. Purchasing seed potato is not as cost effective as the old school method of simply planting your own potato slips, so saving high quality specimens from your potato harvest is the best route to go.
If the current years potatoes were subjected to any blights or disease issues, don't save them. Saving them at this point will only carry the disease issues into your next crop. [Certified seed is not 100% guaranteed to be disease free, but has a lower percentage of disease.]
Once you've harvested your potatoes, those spuds that you select to use for planting need to be stored. The storage period is also a rest period for the potatoes during which they will gradually produce sprouts. The word gradually needs to be stressed - prematurely sprouting potatoes are more apt to rot before it's time to plant them.
To properly store potatoes for seed and prevent Premature sprouting there are several steps that should be taken.
1. Expose them to the light for at least a day - preferably 2 to 3 in order to allow them to turn a little greenish, this will encourage dormancy. [Do not eat potatoes that have turned Green - they are guaranteed to give you a belly ache at the very least.]
2. Brush off any dirt that clings to the potato. Don't wash them - just gently brush off the dirt
3. Place them in a dark, dry and cool location with reasonable air circulation. Higher temperatures encourage premature sprouting and promotes decay. The ideal storage temperature for seed potatoes is 42 - 500 F. A potato should take anywhere from a Month up to 140 days to form sprouts - this varies with different varieties and storage conditions. Although the location needs to be dry in the sense that it is not prone to leaks, flooding, and water accumulation, the air needs to have a suitable moisture content. Potatoes are 75- 80% water, in a dry location for extended periods they will shrivel up and dehydrate - so although the location needs to be dry the air does not have to be and should not be completely dry.
4. Roughly 3 weeks before planting your seed potatoes remove them from storage and place them in a location where they will be exposed to the normal conditions they would experience in nature.
Seed potatoes can be planted whole or cut into halves, or, quarters on larger varieties, with each piece containing an eye or two. Too small of a seed piece will produce a weak plant. They should be cut into pieces which weigh about 1.5 to 2 ounces each and have at least two buds.
Seed potatoes are size coded. B-sized tubers weigh only an ounce or two and do not require cutting. Larger sizes do. See Potato Sizes
Cutting Seed Potatoes
Make the first cut lengthwise in order to divide the cluster of buds at the end of the potato.
Once you've cut the pieces, it is not a good idea to place them directly into the soil. Store the cut pieces for no longer than a week , but a least a full day before planting them in order to allow a dry callus to form. The formation of a callus will minimize the chances of rot. Cut Seed Potato should be stored at room temperature (65 to 70 degrees F) in a well ventilated humid area.