I have one section of my (tiny) yard that receives very little sun and I asked about edibles in this area at a gardening class recently. I was told it was unlikely to produce any food.
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We've planted all kinds of different things in this area over the years. Some things last longer than others, but no food has grown there, yet. Yarrow lasted a few months, but didn't thrive. Beets were planted on the edge of the no-sun-zone and they did okay, but they received *some* sun.
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What has thrived in that area is a white flower that I don't recall the name of...pretty, but not edible.
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My current plan is to use a shepherd's hook in the ground in this area and hang a basket of something. I'm hoping the height will give the plant enough sun to thrive. Does your patio receive any sun up high or along one side or anywhere for awhile during the day?