The survey found that those happiest with their jobs weren't people who spent the majority of the week in the office collaborating with their colleagues and one day a week skipping the commute. Nor were they the people who spent the entire week working from home.

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Instead, the report showed that the most engaged workers were those who spent 60 to 80 percent of their week, or three to four days, working from home, and a minority of their time in the office. Those who spent more or less time working remotely were less enthusiastic about their work, with the lowest numbers occurring among those who spent either all of their time either in the office or at home.

In other words, after untold dollars spent by companies to get workers more committed to their jobs, and study after study suggesting a link between employee engagement and the bottom line, working from home three to four days a week doesn't just make things better for employees. It could also make things better for their boss.

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Unsurprisingly, people feel most plugged in to their jobs when they have some balance -- a little bit of face time and camaraderie at work, and plenty of time to hunker down and get work done from home while avoiding the headaches of going in to the office. What may bewilder many managers, however -- particularly those at companies like Yahoo or Bank of America that scaled back on telecommuting programs in recent years -- is the extent to which that balance is tipped toward working remotely.

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Just four years ago, after all, Gallup found that the "optimal engagement boost" happened for those who worked remotely less than 20 percent of time, a third of where it is now. The 2012 data showed that those who spent more than half their time away from the office were only about as engaged as those who never worked remotely.

Four years on, something has changed dramatically.

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Improvements in technology -- better mobile phones, faster home Internet service, better videoconferencing -- may play some role, though these technologies were also well in place in 2012. Greater numbers of people working from home -- and therefore, more acceptance that could lead employees to feel more comfortable with doing it more -- could also help explain the boost, but the jump isn't profound. Gallup found that in 2016, 43 percent of employees worked remote at least some of the time, compared with 39 percent in 2012.

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Rather, says Jim Harter, chief scientist for workplace management at Gallup, the primary explanation may be that companies are doing more to help remote workers get it right. From handing out technology kits to get people up and running at home to training managers to be clear about job descriptions to building in systems that improve collaboration between those outside the office, he says, companies have been responding to the shift toward remote work with ways to help the work-from-home crowd work. "I think organizations are more intentional about putting resources around it," he said.

While Harter was surprised to see the ideal range for remote working shift so much in four years, he wasn't shocked to see a strong link between employee enthusiasm and working from home. The best predictor for employee engagement, he says, is when workers say they have a "significant amount of time where they get absorbed in their work and time passes quickly," he said. "And when you work remotely, you certainly have more of a chance to get absorbed in your work."

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While flexibility and remote work arrangements were particularly influential on how enthusiastic workers are in their jobs, Gallup's report reminds us there are plenty of other factors, including the struggles those who do have to work in the office cope with every day. In a work world plagued with open office plans, employees who can shut a door on their workspace are 1.3 times more likely to be engaged than other workers, Gallup reports,while those who say they have privacy when they need it are 1.7 times more likely to be engaged.

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Some would even leave their jobs to get those things: About 40 percent of workers would change jobs for a personal office, Gallup found, while one-third said they would leave even for a thermostat set at a comfortable temperature.