Many knitters find their craft a tranquil and even meditative pastime—until knots and tangles in their yarn send them into a fury. But for one group of fanatics, there is nothing more satisfying than a hopelessly tangled web.

Daphne Basnet of Melbourne, Australia, once paid about $50 on eBay for a 25-pound box of snarled yarn, simply for the pleasure of untangling it. “I was so happy, I can’t tell you,” recalls the 58-year-old of her purchase, a mess of about 120 knotted balls.

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