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Since then I've wound the yarn into a ball a couple of times, twisting the yarn in the opposite direction of the spin just enough to un-kink it where it curls over on itself, then work that down towards the end, and so on. The end result turned out rather decent for a first effort, I think:
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In the meantime I'm still working on my nephew's hoodie. I've gotten the front and back done and am almost finished with one sleeve, so we're getting there.
I also gave the hoodie I made for my friends' son to him at his birthday party over the weekend. His parents pointed out that when he's outgrown it, with the neutral colors his little sister can have it afterward, and with their beautiful complexions it will look great on both of them. So it will get plenty of use down the line. Hooray!