“Not a chance,” Ms. Horwood said. “It’s a digital world, and they live in it.”

The Zanders of South Salem, N.Y., experienced a blackout last year, “so we’re getting good at the 1800s in our house,” said Lauren Handel Zander, who runs an executive life-coaching company. Her three children “live for Mommy’s iPad,” she said, likening the first days of the blackout to rehab. “It’s like coming off drugs,” she said. “There’s a 48-hour withdrawal until they’re not asking about the TV every other minute.”

The Zander children did enjoy the unusual undivided attention of a working mom. “Mommy got parked,” Ms. Zander said ruefully. “I’m not as ‘on’ if my kid is attached to one of those devices. I played Clue. I haven’t played Clue in a very long time. We got to hang out more, which was an entire family adjustment, but it’s a good problem to have.”

Among the parents who spoke with pride about newfound family time when their children were forced offline, there were honest admissions about the joy-kill of too much bonding. One 10-year-old boy in Lower Manhattan sweetly told his mother, “This gives us a chance to talk.” After three hours of “and that’s why they need to ditch Sanchez and make Tebow the starter,” she was silently pleading for someone to turn the power on.

“For the first three days, I was full of maternal pride,” said Marjorie Ingall, a writer in the East Village. “’Look at my children: reading by candlelight, cutting out paper dolls, engaged in such brilliant imaginative play. We are so ‘Little House on the Prairie.’ Then Day 3 hit and the charm of screenless togetherness wore off. I was genuinely concerned that we were all going to kill each other.”

By the time the family made their way to a relative’s fully powered home, one of the children “had cracked like an egg, spending three hours glued to the TV and ignoring all humanity,” Ms. Ingall said. “My prediction: It’ll be a week to 10 days before we’re back to all our zoned-out-and-beeping habits.”

William Powers would advise otherwise. “One of greatest skills you can teach a child is: You don’t have to be hooked up to any machine to get through life,” said Mr. Powers, the author of “Hamlet’s BlackBerry: Building a Good Life in the Digital Age.”