The great Greek mathematician Archimedes had the first Eureka! moment 2,200 years ago.

According to legend, inspiration struck Archimedes as he sat in his bathtub, pondering an assignment from his king: Was the royal goldsmith cheating?

The solution, which sent the scholar running naked through the streets shouting Eureka! (I've found it!), arrived in a flash that we call genius.

In a radio documentary on Ideas March 19 at 9 p.m. (9:30 p.m. NT), producer John Chipman explores the origins of Eureka! moments, creativity and the neuroscience that drives them.

In Archimedes' case, the king suspected his goldsmith had mixed in some cheaper metals when he made a new crown, hoarding some of the gold. Could Archimedes prove it — without melting down the crown?

In his tub, Archimedes noticed the water rising. In a flash, he had his solution: displacement.

He could use water displacement to calculate the volume of the crown. Archimedes had already weighed it, so he knew the crown's mass. Divide its mass by its volume, and he'd have its density.

And if king's crown was made with pure gold, it would have the same density as pure gold. It did not, and so Archimedes' test proved the goldsmith was a cheat.

The tale is likely more fiction than fact, but Ideas has some genius tips.

1. Do your homework

There's a reason why Albert Einstein came up with the theory of relativity, and not Albert Brooks: only one of them
was a theoretical physicist.

But intelligence is not the same as creativity. You don't have to be as smart as Einstein to come up with a powerful idea, but you do have to study up on whatever topic you're trying to innovate.

Weird side fact: the actor Albert Brooks was also born Albert Einstein, but changed his last name to avoid confusion
with his more famous namesake.

2. Take a break

Too much focus is bad for creativity. Take a nap, go for a walk, or hop in the bath like Archimedes.

Let your ideas incubate. Taking your mind off the problem is often just what's needed to solve it.

3. Analogies help

Research shows that making the little mental leaps between analogies can unlock larger, more creative Eureka! moments — like Archimedes seeing that his body made the bath water rise, and realizing the crown would do the same thing.

4. Quantity matters

Thomas Edison filed more than 1,000 patents. Pablo Picasso created more than 25,000 works of art.

Chemist Linus Pauling said, "The best way to have a good idea is to have a lot of ideas."

It's good advice.

5. Competition helps

There's a Darwinistic theory of creativity that says the best idea survives after fighting off an army of other ideas.

Brainstorming is good, but only if critical thinking is involved.

Say no to the yes-man!

6. Be prepared to modify

Good ideas become great with constant tinkering, and sometimes that means going back to the drawing board.

That's why it helps to have lots of of ideas. See No. 4

7. Be prepared to verify

 Having a Eureka! moment usually isn't enough. You'll need to do more homework to prove your big idea works.

Don't get discouraged  verification can take years, even decades.

Eureka! moments may happen in a flash, but the entire creative process can take much longer.