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Do potatoes stop growing once the leaves have died back? I would presume so. And what determines the size of a potato? I've had all sizes in one row. And finally, any recommended varieties to get big ones? Although to be honest my Charlotte have been so good this year I'll probably stick with them even though I've not had many big ones.

Yes once the potato leaves and stems are dying back most of the potato growing is done, but I think any remaining goodness and energy they have is sent to the tubers.

Size is due to a variety of factors, among them variety, feeding, growing conditions, moisture and spacing. I always get mixed sizes too and sometimes different from one end of a row to the other. To be honest my poor potatoes always seem to get killed off before their time with Late Blight, so I can't really recommend ones for size. Interested to see what others say though.

Yes I believe they do. The tuber is a store of energy from photosynthesis which can only occur with a fit green leaf. Excess feed results in excess foliage, but starved plants do not produce large tubers. Add in the factor of moisture and you begin to see what the ideal would be.
Warm, bright days but not too hot. Adequate ground moisture, ie within the water table. Steady levels of nitrogen and soils which lead to exploratory roots, being rich, well cultivated but moderately compacted. Deep enough to ensure tubers do not penetrate the surface, even after heavy rain which may wash away the surface, achieved by ridging but with broad flat domes, not mountain peaks! Adequate spacing , minimal competition from weeds and adjacent vegetation.

This year its not been ideal in RG28 , with drier conditions early on, dull cool days, few adequate rain showers, and those that have been received were probably too light and too late to penetrate to the productive depth.
I have noticed that its been necessary to dig deep, as much as a full spit below the planting level (where the "seed" remains broadly intact), to harvest the better spuds. These discovery has led to reworking dug rows to find a "second" level, with some better potatoes, although few in number.

Sarpo Mira, which in my experience, is quite resistant to bight, can produce some huge but somewhat ugly tubers.

ANother vote for Sarpo Mira, it does make big spuds, but their only real use is as a baker and they take somewhat longer to cook than some other varieties... If you want ridiculously big though, find some Kondor, they're not always easily available, but they are the one that those of us that grow for showing as "biggest potato" go for, they have a tendancy to scab and they're nothing to write home about flavourwise, but my word they are big....

Kestrels an Charlotte. 

Agree with Cornykev.........Charlotte always does well, sizes from salad to bakers, Kestrel also a good doer on my clay soil...big tubers even if the weather is pants.

growing spuds here on the coast is a bit of a gamble light sandy soil no rain so i grow earlys in the ground and the rest in sacks it a bit expensive due to compost costs but it does work sarpo and charlotte all seem to suit the way i grow them i also grow nadine which have a great buttery taste those are my familys fav

Jonnyco15, I've lost lots of spuds to slug damage does growing them in bags cure this problem. I'd love to know!