Here's a brain teaser: Your task is to move a single line so that the false arithmetic statement below becomes true.

IV = III + III

Did you get it? In this case, the solution is rather obvious – you should move the first “I” to the right side of the “V,” so that the statement now reads: VI = III + III. Not surprisingly, the vast majority of people (92 percent) quickly solve this problem, as it requires a standard problem-solving approach in which only the answer is altered. What's perhaps a bit more surprising is that nearly 90 percent of patients with brain damage to the prefrontal lobes – this leaves them with severe attentional deficits, unable to control their mental spotlight – are also able to find the answer.

Here’s a much more challenging equation to fix:

III = III + III

In this case, only 43 percent of normal subjects were able to solve the problem. Most stared at the Roman numerals for a few minutes and then surrendered. The patients who couldn’t pay attention, however, had an 82 percent success rate. What accounts for this bizarre result? Why does brain damage dramatically improve performance on a hard creative task? The explanation is rooted in the unexpected nature of the solution, which involves moving the vertical matchstick in the plus sign, transforming it into an equal sign. (The equation is now a simple tautology: III = III = III.) The reason this puzzle is so difficult, at least for people without brain damage, has to do with the standard constraints of math problems. Because we’re not used to thinking about the operator, most people quickly fix their attention on the roman numerals. But that’s a dead end. The patients with a severe cognitive deficit, in contrast, can’t restrict their search. They are forced by their brain injury to consider a much wider range of possible answers. And this is why they’re nearly twice as likely to have a breakthrough.

Of course, this doesn't mean you should take a hammer to your frontal lobes. Being able to direct the spotlight of attention is a crucial talent. However, the creative upside of brain damage – the unexpected benefits of not being able to focus – does reveal something important about the imagination. Sometimes, it helps to consider irrelevant information, to eavesdrop on all the stray associations unfolding in the far reaches of the brain. We are more likely to find the answer because we have less control over where we look.

This helps explain a new study led by Mareike Wieth at Albion College. The scientists surveyed 428 undergrads about their circadian habits, asking them whether they were more productive and alert in the morning or evening. As expected, the overwhelming majority were night owls, which is why they studiously avoided 9 a.m. classes. Then, the scientists gave the students a series of problem-solving tasks. Half of these tasks were creative insight puzzles, in which the answer arrives suddenly and seemingly out of nowhere. Here's a sample insight puzzle:

A man has married 20 women in a small town. All of the women are still alive and none of them are divorced. The man has broken no laws. Who is the man?*

And here's another classic puzzle:

Marsha and Marjorie were born on the same day of the same month of the same year to the same mother and the same father, yet they are not twins. How is that possible?

Did you solve these brain teasers? (The answers are, respectively, priest and triplets.)

The other half of the problems given to the students were standard analytic problems, such as long-division and pre-algebra equations. These questions don't require insights. Instead, they benefit from ordinary focus, as people grind out the answer and check to make sure it's right. The subjects were given four minutes to solve each problem. Half of them were tested early in the morning (8:30 a.m.) and half were tested in the late afternoon (around 5 p.m.).

The results are a testament to the creative virtues of grogginess. When people were tested during their "least optimal time of day" – think of that night owl stumbling into the lab in the early morning – they were significantly more effective at solving insight puzzles. (On one problem, their performance increased by nearly 50 percent.) Performance on the analytic problems, meanwhile, was unaffected by the clock.

The larger lesson is that those sleepy students, like a brain-damaged patient, benefit from the inability to focus. Their minds are drowsy and disorganized, humming with associations that they'd normally ignore. When we need an insight, of course, those stray associations are the source of the answer.

One last piece of evidence: A brand-new study by scientists at the University of Illinois at Chicago compared performance on insight puzzles between sober and drunk students. (They were aiming for real intoxication, giving students enough booze to achieve a blood alcohol level of 0.075.) Once the students achieved "peak intoxication" the scientists gave them a battery of word problems – they're known as remote associate tests – that are often solved in a moment of insight. Here's a sample problem. Your task is to find the one additional word that goes with the following triad of words:

Cracker Union Rabbit

In this case, the answer is "jack." According to the data, drunk students solved more of these word problems in less time. They also were much more likely to perceive their solutions as the result of a sudden insight. And the differences were dramatic: The alcohol made subjects nearly 30 percent more likely to find the unexpected solution.

Once again, the explanation for this effect returns us to the benefits of not being able to pay attention. The stupor of alcohol, like the haze of the early morning, makes it harder for us to ignore those unlikely thoughts and remote associations that are such important elements of the imagination. So the next time you are in need of insight, avoid caffeine and concentration. Don't chain yourself to your desk. Instead, set the alarm a few minutes early and wallow in your groggy thoughts. And if that doesn't work, chug a beer.

Image: Rachel Carter/Flickr