Friday, September 17, 2010

An Unusual Series

Which may not be all that interesting to you, unless you're interested in recreational math. For lots of you, that may be sort of an oxymoron. (Although, I'm hoping it's less likely among readers of my blog than it would be among the general population.)

Here's the idea. Start with an integer. Add its digits together. If that sum is even, halve the number (not the sum of digits) to get the next number. If the sum is odd, add one to the number.

For instance, suppose we start with the number 10. Its digits sum to 1+0 = 1, so we add 1 to get 11. Those digits sum to 1+1 = 2, so we halve it to get 11/2 = 5.5. Those digits sum up to 5+5 = 10, so we again halve the number to get 2.75. Those digits sum up to 2+7+5 = 14, so we again halve the number to get 1.375...well, I think you get the idea.

On the other hand, suppose you start out with the number 1. Its one digit sums to 1, so we add 1 to get 2. Its single digit sums to 2, so we halve it to get 1 again. Obviously, this series repeats forever: 1, 2, 1, 2, 1, etc.

The first eight numbers, 1 through 8, all end up at that same repeating sequence. The next number, 9, leads immediately to 10, which starts out as I worked out above, and then goes on indefinitely: Each number has one more digit after the decimal point than the preceding number, so the series never repeats, and it never reaches zero, either.

In my limited trials, every integer I've started out with either ends up with the repeating sequence 1, 2, 1, 2, 1, ..., or else it eventually merges with the same series that you get if you start with 10 (or 9, for that matter). So, two questions for those of you who might like to play with this kind of thing:
  1. Is it true that the series for any integer always either ends with the sequence 1, 2, 1, 2, 1, ..., or else merges with the series that starts with 10?
  2. Consider the series that starts with 10. As we said, it goes on forever, without repeating. What is the average of the numbers in that infinite series?
Neither of these questions can be answered definitively (as far as I can tell) with brute-force computation, although the results might be suggestive. If you do want to try some computations, use an infinite-precision package; our friend Bernie has already tried it with ordinary floating-point numbers (eight-byte doubles, I think), and roundoff error rendered everything after about the 15th number quickly invalid.

P.S. Don't ask me how I got started thinking about the series. It's inspired in part by this guy, but I've already forgotten how I decided to think about this variant.

Friday, September 3, 2010

Grasping at Genius

No, this isn't about me trying to become a genius. My aim is a lot more modest: trying to draw a bead on what genius is. Partly this is motivated by my last post about music, but mostly it came out of a discussion I had several years ago with a co-worker over whether athletes could be geniuses at their sport. I thought they could, and he thought not. He conceded that they had some outstanding skill, but felt that it would be demeaning the word "genius" to call it that. I was willing to be a bit more expansive with the term. One does have to be a little careful—probably half the parents out there think their precious little ones are geniuses—but limiting genius to a specified list of fields seemed unnecessarily restrictive to me.

The discussion more or less had to end there because we never really grappled with the larger issue of what genius really is, and without that any debate over whether it means anything in sports is putting the cart before the horse. I want to tackle that now, so I can go back and win the original argument.

First of all—because I'm sick and tired of hearing about it, even now—what is genius not? It is not a high IQ, or intelligence quotient. Lots of folks are intimidated by numbers (especially, but not exclusively, those who do not feel comfortable around them), to the point that any description using them feels more objective and unassailable. Well, they might be that, but what's lost when a number is attached to anything is the process by which that number was derived. If you don't know and understand that process, the number—while not exactly meaningless—is not as reliable as it sounds.

In the case of IQ, the formula is generally straightforward; what's not so clear are the principles on which questions are selected for IQ tests. If you've ever taken one, you know that questions on such tests are fairly narrowly circumscribed: which one of these things doesn't belong, how many blocks are there, numerical or word analogies, etc. The only thing that we can be sure IQ tests measure is how well someone takes IQ tests. Beyond that patently circular assertion, it gets hazy. Does it measure intelligence? How about genius? There are lots of folks who have very high IQs (Marilyn vos Savant—really? that kind of name?—comes to mind) who nonetheless evince no obvious signs of genius. To her credit, vos Savant doesn't make any claims of genius for herself.

If we can't rely on a test to identify genius, we are back to Potter Stewart's famous dictum (in his concurring opinion in Jacobellis v. Ohio regarding hard-core pornography): "I know it when I see it." So where do we see it?

If we start with the so-called hard sciences (physics and chemistry), plus mathematics, I think you'll find little argument that folks like Archimedes, Isaac Newton, Carl Friedrich Gauss, and Albert Einstein were geniuses. Expand that to all of letters and sciences, and you embrace other noted geniuses, such as Charles Darwin, Louis Pasteur, and B.F. Skinner. But maybe these get a little dicier. These are great scientists, to be sure, but what about them promotes them beyond the ordinary rabble?

You might expect that things would get dicier still when we go to the fine arts, but at least in my experience I find less argument about ascribing genius to artists like Leonardo da Vinci (also an engineer), William Shakespeare, Auguste Rodin. How about musicians? Ludwig van Beethoven, Richard Wagner, and Igor Stravinsky all wear the mantle of genius, and wear it rather comfortably at that. (Yes, I realize these are all dead white dudes. I'll get to that in a moment.)

Let's pause a while and take stock of what we have. Accepting for the sake of discussion that these people are all geniuses, what makes them so? They don't just do what ordinary people in their professions do, only better—although by and large, they do do those things better. They also don't just do what ordinary people can't do—although, again, they do do that, too. What sets them apart is that they do things that ordinary people in their profession could never even conceive of, before the geniuses did. Arthur Schopenhauer put it this way:
"Talent hits a target no one else can hit; genius hits a target no one else can see."
I must emphasize that innovation is a vital part of this. One of Newton's most important contributions to physics was a mathematical demonstration of the law of universal gravitation (the so-called "inverse square law" of gravitation) from Kepler's observations and laws of planetary orbits. That same law is derived countless times over by students in undergraduate physics classes around the world (albeit using analysis, rather than the essentially geometrical means that Newton employed). That doesn't mean that any of them, let alone each of them, is a budding Newton, for likely none of them, plucked at birth and set down in a pre-Newtonian world, could have done what Newton did. Newton's genius lay in blazing the trail that future scientists and students would follow.

In that context, then, let me add a few other names to the list: Charlie Parker, Miles Davis, Herbie Hancock. Jazz is an art form, among others, that combines composition and performance in a single moment, adding for the first time—to my list, anyway—the element of dynamism. (I don't mean to slight other performance geniuses, such as actors and stand-up comedians, but I'm trying to make a point!) Although jazz tunes are composed to a certain extent, a fundamental aspect of jazz performance is improvisation. No two jazz performances are ever exactly the same—not, at any rate, to the extent that classical music performances are alike. The music is constantly written and rewritten by each new performer that approaches it, and each new performer must contend not only with the structure of the music, but with the performers around him or her, in an endeavor that is, in the best of cases, at once collaborative and competitive. And genius denotes the ability, moment to moment, to conceive and perform what others in that situation could not even imagine.

From that point, how far of a step can it be to arrive at sports? I'm going to talk about basketball, because it's the sport with which I'm most familiar, but similar arguments could be made for other sports. (Imagine, for instance, the shots that Tiger Woods can execute that others would never even attempt, or the sudden volley, deft but fierce, of Pete Sampras.) Basketball, like jazz, requires the constant attention of the athlete to the ever-changing state of the game, from the highest level down to the smallest detail, and the ability to respond to that state, all on the spur of the moment. Where's that pick going to be in five seconds? What are the possible tactical options available to me, given the current score and time remaining? Seeing the passing lane halfway down the court is a geometric exercise in negotiating tangled world-lines in the four dimensions of space and time; to actually complete the pass, when everyone else is watching, one must summon the legerdemain of a practiced conjurer.

We think of sports as an essentially physical activity (which is probably why my co-worker could never attach the genius label to an athlete), but in its own way it is as demanding on the intellect as the most abstruse mathematical theorem, and unlike the mathematicians, who can return now and again to their labors when it suits them, the athlete has only the splittiest of split-seconds to act—or else the instant is gone. Who are we to say that genius could not act here, as well as anywhere else?

We may debate whether or not Wilt Chamberlain, Michael Jordan, or Magic Johnson merit the label of genius, whether or not what they do exceeds the conception of their colleagues. But not, in my opinion, whether the question makes sense. Even we non-geniuses can see that, I think.