When the geekosphere exploded in May with the story of Yitang Zhang, the unknown University of New Hampshire lecturer who single-handedly solved one of the oldest problems in mathematics, we were intrigued. We loved his grunt-to-genius rise from Subway sandwich maker to academic superbrain. We appreciated the sheer complexity of the twin primes problem. But, mostly, we marveled at this description of how Zhang finally cracked the puzzle:

"Without communicating with the field's experts, Zhang started thinking about the problem," writes Erica Klarreich (@EricaKlarreich) in Simons Science News. "After three years, however, he had made no progress. 'I was so tired,' he said. To take a break, Zhang visited a friend in Colorado last summer. There, on July 3, during a half-hour lull in his friend's backyard before leaving for a concert, the solution suddenly came to him. 'I immediately realized that it would work,' he said."

What is so revolutionary about Zhang's process? It flies in the face of everything corporate America believes and practices in the pursuit of groundbreaking innovation. We submit:

1. Zhang worked alone. No brainstorming, no collaboration, no teamwork.

2. His solution did not come in a flash of inspiration. According to Wired, "Zhang achieved his result not via a radically new approach to the problem, but by applying existing methods with great perseverance."

3. The defining moment, or his shift in thinking, occurred during 30 minutes of pure, unadulterated downtime.

The third point reminds us of research findings we've read before in publications including WiredHarvard Business Review, and Psychology Today: Time away from work actually improves the quality of that work.

The benefits of downtime are not new to us. Yet a Harris Interactive study in 2011 found that 57 percent of Americans left the majority of their vacation days on the table that year. Why is it that so many of us have such a hard time shutting down and checking out?

It could be psychological.

Studies cited by Hal E. Hershfield in Psychology Today show that people like to keep busy. If given the choice of sitting idly or walking for 15 minutes, we almost always choose to move. Even when someone forces busyness upon us, we are happier for it than if we'd just sat still instead.

Today, our need for busyness manifests itself scheduling more meetings, agreeing to more business travel, or relentlessly IMing colleagues three offices away. As our smartphones and 24/7 culture feed and enable our habits, we are denying our brains the very thing they need to generate brilliant ideas: downtime.

One solution is to simply take all of our vacation days.

Another is to build brain breaks into our daily schedules. That's what Jeff Weiner (@jeffweiner), CEO of LinkedIn, does. In the blog post "The Importance of Scheduling Nothing," he explains:

"If you were to see my calendar, you'd probably notice a host of time slots grayed out but with no indication of what's going on... the gray sections reflect 'buffers,' or time periods I've purposely kept clear of meetings.

"In aggregate, I schedule between 90 minutes and two hours of these buffers every day (broken down into 30- to 90-minute blocks). It's a system I developed over the last several years in response to a schedule that was becoming so jammed with back-to-back meetings that I had little time left to process what was going on around me or [to] just think.

"At first, these buffers felt like indulgences. I could have been using the time to catch up on meetings I had pushed out or said 'no' to. But over time I realized not only were these breaks important, they were absolutely necessary in order for me to do my job."

Why is that? "As the company grows larger," he continues, "as the breadth and depth of your initiatives expand, and as the competitive and technological landscapes continue to shift at an accelerating rate, you will require more time than ever before to just think--think about what the company will look like in three to five years; think about the best way to improve an already popular product or [to] address an unmet customer need; think about how you can widen a competitive advantage or close a competitive gap, etc.

"That thinking, if done properly, requires uninterrupted focus . . . and that time will only be available if you carve it out for yourself. Conversely, if you don't take the time to think proactively, you will increasingly find yourself reacting to your environment rather than influencing it. The resulting situation will inevitably require far more time (and meetings) than thinking strategically would have to begin with."