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Potatoes are easy to grow, and great fun even in a small garden
The principle is you simply plant one tuber and a couple of months later you get dozens. Any variety will do fine in pots but it makes sense to grow earlies because the will crop quickly and they'll avoid the possibility of potato blight.
As the weather warms up they'll grow amazingly quickly, make sure they are in full sun if possible and water daily.
By late June or early July your potatoes should be ready. Have a little feel to check they are a decent size, if they are just tip the whole lot out on a sheet of plastic and gather the bounty. Pur spent compost on the garden or re-invigorate with John Innes base and use again.
Planting tips
Use proper 'seed' tubers rather than eaters that have gone sprouty. Good results are much more likely if you start with virus free seed spuds.
Try and find somewhere that you can buy just the number of tubers you need - don't be tempted to plant too many in your containers
Potato varieties
Earlies
The best choice for growing in pots are early varieties, which produce eating-sized tubers in early summer. They generally have less top growth, so make neater container subjects. Try: 'Accent' 'Lady Christl' 'Rocket' 'Swift' 'Red Duke of York'
Baby earlies, such as the red-skinned salad variety 'Mimi' have been bred to produce lots of cherry-tomato-sized tubers for cooking and eating whole. A pot a great place to grow such tiny tubers.
Salad potatoes
Second earlies crop a few weeks later than earlies. Many are salad varieties, which have a waxy texture and are good boiled to eat hot or cold. Try: 'Maris Peer' 'Charlotte' 'Roseval' 'Anya' (an interesting newcomer - a cross 'Désirée' and tasty 'Pink Fir Apple'.
Later varieties
Don't bother with the late ones, they'll get blight
Potato blight
Grow Sarpo varieties to avoid blight - Sarpo 'Axona' is a good one.
Keep the foliage dry by watering the compost only.
Pick off infected leaves.
When a plant is badly affected, cut off the top growth and burn or bin it - don't compost it. If the tubers are unaffected, they can be eaten.
Grow early varieties - these can be started early and harvested before mid-July when blight becomes a problem.