Brain teasers

A cowboy rides into town on Friday, stays two consecutive days, and leaves on Friday. How could that be?

That's a typical interview question asked by cool companies that seek creative thinkers. The best candidates don't score points for the right answer, just for the right approach, kind of like new math.

Google Labs even had its own aptitude test, called GLAT. My favorite question on that test was, "what was the most beautiful math question ever derived?"

So what is the deal with the cowboy? If Friday is a day of the week, for which our thinking is reinforced by the fact that the cowboy stayed two consecutive days, the question seems impossible. So our assumption that Friday is a day of the week could be wrong. For example, Friday could be the name of the cowboy's horse.

Companies have come a long way from, "tell me about your last job." But they still ask you to work on Friday.

The search for beauty

The most beautiful woman, man, science breakthrough, literature, and so on usually merits a prize:
Miss America, Mr. Universe, the Nobel prize, the Pulitzer prize.

The men and women are easy, just ask People Magazine: J Lo, Angelina, Brad Pitt, Tom Cruise. Science breakthrough? That's pretty easy too: Einstein's Theory of Relativity and the discovery of DNA would be high on anyone's list. Most beautiful poem? Shall I compare thee to a summer's day?

What do musicians say when asked for the most beautiful piece of music ever written? My informal survey says Schubert's Cello Quintet in C gets the prize.

And how about Google candidates when asked for the most beautiful math equation ever derived? e to the i pi equals minus 1 comes to mind. Just think: you put together an imaginary number and two transcendental numbers and get minus 1 as the result. This is at least as cool as Pythagoras. Just beautiful.

Google and googol

Incorporated nine years ago, Google’s market cap is now $165 billion, seven times more than just three years ago, when it went public. Still, $20 million for sending a traveling rover to the moon, which is what Google offered last week, is a big number.

But it’s a lot smaller than the number “googol”, spelled googol as opposed to Google for the company name, which is defined as the number one followed by one hundred zeroes. The term was invented by mathematician Edward Kasner’s nephew in 1920.

A googol is larger than the number of atoms in the observable universe, and it is estimated that a supermassive black hole will need a googol years to evaporate from Hawking radiation. A googol is slightly less than 70 factorial, which means that there are more than a googol ways to order just 70 people or objects.

The company’s headquarters is known as the Googleplex. Googolplex – with an ol and not an le – is also a number: it’s a one followed by googol zeroes. Carl Sagan wrote that it would be impossible to write the number googolplex because even using 1 point type it would take more space than exists in the universe.

That $20 million Google Lunar X Prize? It’s unlikely to be won. You only have a little over five years to win it. And with the way gas prices have been going, the money may not even cover the fuel cost.

The hype behind the iPhone

On Friday at 6 p.m. local time, Apple will begin selling its iPhone to the public. The hype is impressive: Google shows over ninety million hits for Apple iPhone. Over ten thousand print articles have covered it. More than 48 hours ahead of the launch, customers are already lining up at some Apple stores, like the one on Fifth Avenue in New York.

Is this unusual? How often does a product like this come along?

Hype is never in short supply when it comes to new product introductions. In 1981, RCA announced that, with sales of just 52,000 units in five weeks, its VideoDisc system achieved the most successful introduction of any major electronic product in history. Right.

To reach the fifty million user benchmark it took the telephone 43 years, radio 38 years, TV 13 years, cable 10 years, and the Internet just 5 years. But some individual products blow this away: the Sony Walkman sold 300 million units in the 80s and 90s. And Apple’s iPod is on a steeper growth curve than the Walkman.

Here’s the math: when a hot new product or technology or even idea takes hold, for a certain period of time the growth is proportional to the number of units sold. For example, if 10,000 units have been sold, the growth rate is, say, 5,000. When the number of units grows to 20,000, the growth rate at that point is 10,000, and so on. The more units sold, the higher the growth rate.

Mathematicians call such growth “e to the x”. Obviously this can’t go on forever. At some point the growth rate slows, levels off, and declines as the product or idea matures and is then replaced by the next generation. But until then it’s a fun ride for the manufacturers and the customers, too.

One thing is for sure. With the introduction of the Macintosh computer in 1984, the iPod in 2001, and now the iPhone in 2007, Apple has a knack for going e to the x.