Review: Ready Player One

Ernest Cline's Ready Player One (Crown Publishing, 2011) features no grand, sweeping philosophical statements, no startling revelation about human nature, no moral judgments or object lessons.  Like the game that forms the backbone of its plot, it is an adventure with a beginning, a middle, and an end, and it rests its case on that straightforward simplicity.

Wade Watts, like the protagonist of many a science fiction novel, is a high-school student-cum-computer-geek in a dystopian society, but unlike many a science fiction novel, RP1 hardly dwells a second on the dystopia part.  The world of 2044 is in ruins, due to a catastrophic shortfall in fossil fuels, but this crisis is put in primarily to motivate the near-universal emotional investment in OASIS (Ontologically Anthropocentric Sensory Immersive Simulation), a sort of virtual-reality massively multiplayer online game that serves simultaneously as school, work, and escape for most citizens, including the continually impoverished Wade.

OASIS was the brainchild of James Halliday, a reclusive, Wozniakian genius with an intense penchant for 1980s pop culture, who, with his more affable business partner, Ogden Morrow, built a multi-billion dollar computer game empire starting in the 1990s.  Over the years, they gradually drifted apart, as Morrow focused on sustaining their company after the death of his wife (his and James's childhood friend) and Halliday seemed to fall deeper into mental illness.

The events of RP1 are set in motion when Halliday dies in 2039.  His death is announced not on the obituary page, but in a video will and testament shot by Halliday himself.  Halliday had no wife, no children, no surviving relatives at all, so in his video, he explains that he will bequeath his entire estate (valued at about a quarter of a trillion dollars) to the first person to solve a series of puzzles embedded into OASIS and its millions of fictional worlds.

Wade had a boundless admiration for Halliday even before his death, so he knows all about 1980s culture, an asset that will stand him in good stead in his quest for the billions.  The problem is, so do many of the other OASIS users, including his best friend Aech (pronounced "H"), geek-girl blogger Art3mis, and the obligatory bad guys, the faceless multi-national corporation Innovative Online Industries (or IOI)—none of whom Wade has actually met face-to-face.  Throughout RP1, Wade will have to contend with each of them and his other rivals, some of who are ready and willing to commit murder and worse in their race for the prize, as well as Halliday's own devilish imagination and his obsession with the 1980s. 

RP1 is written in a quick, breezy style with pulpish overtones.  (Of course, for those who grew up with the golden age of science fiction, the pulp might be a positive.)  In developing his story, Cline feels compelled to explain a bit of Halliday's world creation, and as a result occasionally gets caught up in his own world creation.  From time to time, we are treated with technical details on how his characters connect to OASIS—details that will abruptly jar many readers from his otherwise breathlessly scripted (and somewhat thin) plot.

The 1980s pop culture references are another matter.  They are dotted liberally throughout the book, sometimes merely for flavor, other times integral to the plot.  And Cline's novel features a slightly implausible ending, albeit one that mirrors that of popular movies in its time frame.  For those of us who grew up in the 1980s, reading RP1 is a bit like watching retrospective "clip" episodes of shows like Silver Spoons and Family Ties (one of Halliday's favorites); others will just be bemused by the constant parade of cultural touchstones they have no connection with. 

To a certain extent, Cline is trying to maintain his footing on a slippery slope.  The technical and pop-culture references are consistent enough to suggest that he had more in his back pocket, details that would have appealed to a very specific audience, but which he held back in aiming for a broader audience.  On the other hand, if he had held back even more, RP1 might have been more accessible, but it would have lost much of its childlike appeal.

Ultimately, RP1 spreads it on thick with its geek and pop culture details, thick enough to turn off readers who don't sympathize with its emphasis.  One gets the distinct impression, however, that it otherwise wouldn't be substantial enough to satisfy those of us who do, and while it teeters on the precipice from time to time, RP1 just does get the job done.

Brian's 0-10 score: 6.0

Sense and Mind

One of the staples of parapsychology is extrasensory perception—ESP.  Apparently, in the early days of ESP investigations, a standard deck of cards was used, but it had some infelicities: The cards had in some cases complex designs that some claimed would interfere with the measuring of ESP ability, and the backs could be used by charlatans to identify cards by means other than honest ESP.  Thus were born the Zener cards.

Zener cards are those specially designed ESP cards that you've no doubt seen: a circle, a plus, wavy lines, a square, and a star—five of each in a 25-card deck.

OHAI BACON.

 

Not only are they simple, straightforward designs, but when placed in the foregoing order, they also embody (in some intuitive way) the numbers one, two, three, four, and five.  Nonetheless, when they were first introduced, Zener cards had many of the same problems as did the ordinary playing cards.  The first Zener cards were made of thin enough paper that it soon became evident that some purported ESPers were simply looking through the cards.  They were subsequently made with thicker paper with opaque backs.

Another problem that arose was that some of the first ESP experiments allowed the participants to see the cards as they were guessed.  In a 25-card deck, random guessing should permit you to correctly guess one-fifth of the cards, or five of them.

However, if you are permitted to see each card after guessing, you can determine which pattern is most likely to show up on the next card.  For instance, suppose you guess that the first card is a circle.  It comes up, let us say, a square.  That one is guessed wrong, but by seeing that the first card is actually a square, you gain the knowledge that the second card is slightly less likely to be a square than any of the other patterns (since there are only four squares left, but five of each of the others).  Each succeeding card gives you even more information.  By the end, with careful counting, the last card is precisely determined; it is the only one that has only shown up four times.

If you always guess optimally, you will correctly guess almost nine cards out of the 25.  That is a level of accuracy that one would otherwise obtain with a probability of only 0.05—the level at which one is provisionally determined to possess genuine ESP.  Needless to say, such experiments were quickly barred.

Suppose, though, that you did an ESP exhibition.  You are not permitted to see the cards after each guess, but you do get to hear the response of the audience to each successive guess and card.  Even without collusion, it isn't much of a stretch to imagine that you'd be able to determine whether you guessed correctly or not.  How many will you guess correctly now, on average?  It should be clear that the number should be somewhere between five and nine, since you have more information than when you didn't get any feedback, but somewhat less information than when you saw each card.

To see some of the issues in determining the expectation, consider a much shorter deck: a five-card deck, with one of each design.  If you receive no feedback at all, you should be able to guess each card correctly with probability 1/5, or an average of one correct card in all.

On the other hand, suppose you see each card after you guess it.  The first card, you guess correctly with probability 1/5.  Having seen what card it actually is, you know which four cards remain, so you guess the second card with probability 1/4.  The third card is guessed with probability 1/3, the second card with probability 1/2, and the last card with complete certainty—probability 1.  To determine the average number of correct cards, simply add up the probabilities: 1/5 + 1/4 + 1/3 + 1/2 + 1 = 137/60; a bit more than two-and-a-quarter cards.

Now, suppose you find out only if your guess was correct.  For the first card, you have no reason to expect that any pattern is more likely than any other; therefore, without loss of generality, suppose that you guess a circle.  You will be correct with probability 1/5.

Suppose you guess correctly.  You are left with a four-card deck, each card equally likely.  Again, since you have no reason to believe any pattern is more likely to be in the second position than any other, you can guess any pattern; let us suppose you guess a plus sign.  This time, you will be correct with probability 1/4.

On the other hand, suppose your guess of a circle on the first card was incorrect.  In that case, the circle must be somewhere in the four remaining cards; it is 1/4 likely to be in the second position.  The other patterns, however, could still have been in any of the five original positions, including the first one.  You know only that it was not a circle.  Therefore, you should also guess that the second card is a circle; we are once again correct with probability 1/4.

It is with the third card that matters become more interesting.  Suppose you guessed correctly on both of the first two cards: the first was a circle, and the second was a plus sign.  You can guess any of the three remaining patterns for the third card; let us suppose that you guess the wavy lines.  You will be correct with probability 1/3.

Or, if you guessed both cards wrong—neither of the first two cards was a circle—you should guess a circle once again for the third card, which will be correct with probability 1/3 again.

Or, if you guessed right on the first card but wrong on the second—that is, if the first card was a circle, but the second card was not a plus sign—you should guess a plus sign again on the third card, which will be correct with probability 1/3 yet again.

The last case is the difference.  If you guessed wrong on the first card, but right on the second, then we know only that the second card was a circle.  The other remaining cards could have been any of the four remaining patterns with equal likelihood.  You can guess any of them, but you can do no better than a probability of 1/4 of guessing correctly.

The analyses of the fourth and fifth cards are more complex.  So the question of this post is: What is a better way of approaching the problem?  What is the average number of cards you will guess correctly?

EDIT: The second question of this post is: Suppose you have k cards, all distinct.  If you always guess optimally, and find out whether each guess was correct (but not what the card actually was), then what is the expected number of correct guesses?  Does this number approach a limit as k increases without bound?  If so, what is that limit?  The answer may surprise you. 

(If you are not given any feedback on your guess, then the expected number of correct guesses is always k × 1/k = 1.  If you get to see each card after you guess it, the expected number of correct guesses is 1 + 1/2 + 1/3 + ··· + 1/k, which goes up as the natural log of k (and is therefore unbounded).  The question above has to do with the situation where you only get feedback on whether your guess was correct or not.)

The Limitations of Sense

As I've mentioned previously, I lived in the dorms in college.  In addition to balky vending machines, the dorms also had a number of lounges—common areas on selected floors for people to gather for the purpose of studying (if they didn't mind a bit of noise), watching TV, or generally screwing around.  And, from time to time, there was the occasional Bible study group.

I hasten to emphasize that the study group people (who generally lived in the dorms themselves) were very reasonable about their use of the lounge.  They were perfectly willing to wander around in search of a mostly unused lounge, and they asked the others instead of just plopping themselves down and using the space.  In my own turn, I was perfectly willing to move over to defrag the chairspace in the lounge and allow them their own section.

Once, though, they did manage to irritate me.

I had settled in with my Walkman, listening to an album.  (For the benefit of those of you who were born in this millennium: Songs used to be sold on physical media, called "vinyl" or "records."  These records could be "singles," or they could be multiple songs sold on one "album."  We had this innovation—developed by Sony, a company that existed even then—called a Walkman, which played "tapes," on which songs could be transferred from the record.  It was called a Walkman because you could walk around with it.  You could listen to a whole entire album and not be tethered to your "component stereo system," which was a collection of devices used to play music at a time when computers had memory sizes measured in kilobytes.  We thought it was great.)

Anyway, the Bible group came in and said they wanted to use the lounge and they promised not to be too loud.  Since I was the only other one in the room and I didn't want to be a complete jackass, I cheerfully agreed and moved over to the other side.  But in doing so, I took off my headphones.  And so, as they began discussing the Bible, I listened to them.  It was interesting, after all.

After some time, however, I guess it became increasingly evident that I was listening to them, and since it was apparently one of their objectives to spread the word to as many people as they could, they began working on me.  Now, I was brought up without any religious background.  (Oddly, I do recall that we had a napkin holder that had some strange incantation on it about "daily bread," although that was never explained to me.  I had to find out about it on my own.  But that's a story for another time.  Essentially, there was no religion in my upbringing, at all.)

What's more, I had by this time become fascinated by science, and the scientific method.  I didn't have a firm idea, perhaps, of how science got done, exactly, but I did have the notion that people were fallible, and experiments were conducted so that we could find things out without relying solely on fallible humans.  And it seemed to me that the more fantastical stories in the Bible (as opposed to the moral precepts, say) simply would not stand up to any kind of scientific inquiry.  I did not believe that there existed anything like the Christian god.  And I'm sorry to say that, somehow, that came out.

Well, the floodgates opened up after that.  And I just could not get them closed back up.  For some reason, I was made to answer for the slightest failing or shortcoming of science as it pertained to anything, and I mean anything, in the Bible.  To be sure, I was not blameless in this; at that age, I had not learned to adopt the sort of detached self-doubt that I can effect these days, and I was unfoundedly certain about the points I made, which landed me in some hot water.

I don't remember how I managed to extricate myself from the "discussion," but I do know that it took a couple of hours, after which I went to my room and lay down.  I was exhausted.

A few of them came up to me the next day, and apologized for their aggressiveness.  I said I understood, and apologized for my unseemly certainty.  But it set me to thinking: I did feel pretty certain about my atheism.  Why?  What made me feel so certain?  I had some vague sense that it had something to do with a kind of epistemological conservatism (though I wouldn't have known to put it in such a way)—the idea that one believes in as few things as is possible to understand the world—and the proposition that extraordinary claims require extraordinary evidence.

It took me some years, however, before I could fully work out what my situation was with regard to atheism, and agnosticism, and all that.  It came about like this:

Much later, I was talking to this fellow, and I mentioned some of this mess I got in with the Bible study group.  And so he asked me, what did make me so certain?  He thought that people who could feel so certain that there was no god were just as scientifically irresponsible as those who could feel certain that there was one.

Fortunately, by this time, I had read Wittgenstein (I'll bet that's the only time you'll hear anyone consider it fortunate to have read Wittgenstein, and by the way, he looks just about that crazy in every picture of him I've ever seen), and I knew he had, too, so I could express it a bit more concisely.  I said that I was about as certain that there was no god as Wittgenstein was that he had a hand.  What good ol' Wittgenstein—and I, by extension—meant by that was that the knowledge that one has a hand represents an upper limit of certainty: a limit imposed by our senses.  We know it not because it is logically proven beyond a shadow of a doubt, but because doubt itself is pointless in this regard.  In other words, the degree to which we know it is a milestone of certainty—in a very real sense, defines it.  In fact, I think Wittgenstein says as much, right at the very start of his final work, On Certainty:

If you do know that here is one hand, we'll grant you all the rest.

My friend was satisfied by that, I believe, and he walked away.  As he walked out, though it hit me that that was it—that the limitations of my senses were the basis of my "certainty" that there was no god.

To begin with: From time to time, some atheist wag will remark that we have no more evidence for the existence of the Christian god than we do for, say, the Flying Spaghetti Monster.  Which is true, so far as it goes, but it doesn't really establish atheism (the belief that there is no god) as it does agnosticism (the lack of a belief that there is a god).

So then, the hypothetical line of questioning goes, what would it take to establish the existence of a god in any kind of scientific way?  Because, as I tell others, if you take a position against something, then as a self-check, you must ask yourself what it would take to convince yourself you were wrong.  Because if there's no amount of evidence that would do it, then your position isn't a scientific one; it can't be falsified.

I thought about all the miracles that are said to be the work of some god or another, all the things that happened that could not be explained.  In most cases, I rather thought that these were evidence less for a god than for the selective ingenuity of humans: If people wanted to believe in something, they were remarkably ingenious about how they managed to assemble evidence in its favor.  But if they didn't want to believe it, that ingenuity mysteriously went away.  In other cases, I couldn't come up with a plausible explanation, except to say that the people who related these stories (thousands of years ago, remember) were either mistaken or, possibly, exaggerating.  That might not have satisfied anyone who was truly on the fence, but it satisfied me.

It boiled down, therefore, to what I could personally witness that would convince me I was wrong.  What could a supernatural being do that would sway me?  It quickly occurred to me that whatever evidence could possibly support the claim to existence of a god had to be much more extraordinary than the possibility that my senses were fallible.  When it came to the existence of a god, I could not grant that I had a hand.

We hear "Seeing is believing," but we see things all the time that, it later turns out, aren't true.  And so, not as an expression of any desire, but simply as an acknowledgement that my senses can fail, catastrophically at times, I flatly admit an incapacity to believe in a god, any god (as normally represented—I obviously don't mean just a super-powerful being, but someone who brought about the world).  It's a personal incapacity, not one that I could possibly extend to anyone else, but it's insuperable just the same.

If It's Negative in Area, Do I Get a Refund for Buying It?

I realize this is mostly crazy on my part, but honestly, I really wish real estate people would stop using the plus-minus sign (±) in this jackass way.

The Tip of the Iceberg

A couple of weeks ago, as I write this, Dharun Ravi was found guilty of invasion of privacy and a host of other charges in a sequence of incidents, including spying via webcam, that ultimately culminated in the suicide of his roommate Tyler Clementi (left).  Ravi faces up to ten years' imprisonment, and deportation to his native India.

Now, since it's been a couple of weeks, a lot has already been written about whether or not Ravi was culpable, whether others had a role, what it says about us as a society that we continue to demonize and ridicule homosexuality (or conversely, what it says about us that we are able to demonize and ridicule someone for being a peeping Tom and a loudmouth).  I'm not going to say anything about that.  As is my wont, I'm going to talk about statistics, but with an eye toward how we perceive events like this.

In a way, those who wonder how we can hound Ravi the way we do have a point, even if I disagree with their larger perspective: What Ravi did, as wrong as it was, is probably happening all over the country—or the world—as we speak.  Is Ravi wronger because what he did led to Clementi's suicide?  Should he, in effect, be the scapegoat on which we place all the otherwise indistinguishable wrongs that, by sheer dumb luck, resulted in nothing more than a change of roommates?  I've been following the Ravi/Clementi case for a few months, after Clementi's suicide but before the trial began, and I seem to recall that Clementi did in fact look into switching rooms, but for whatever reason did not manage to do so before his death.  If he had changed rooms, where would we be now?  Would we be up in arms about homophobia and scapegoating?

This is only part of a general problem that human beings have with assessing rare events.  To be sure, it's not simply a matter of placing too great an emphasis on the result of those events, although we do do that.  (Many of us greatly fear the rare airplane crashes, even though they are at least an order of magnitude safer than road travel by practically any metric you care to choose.)  More than that, it's that we just do not have the vocabulary to compare these rare events, and their consequences, with their more typical brethren.

Interestingly, we don't really run into significant roadblocks with their opposite number, the rare non-events.  If someone intentionally shoots a bullet into a crowd, and against incredible odds, manages to hit no one at all, we still find them guilty of reckless endangerment.  The rare non-homicide doesn't conceal from us from the essential wrongness of the act.

But Ravi's case, and others like it, put us in a quandary.  Despite what others have said, I don't believe what Ravi (right) did led inevitably to Clementi's suicide.  We tend to think so because Clementi did in fact die, and what Ravi did is reprehensible and did in fact lead materially to Clementi's death.  But to think that it was the unavoidable outcome of what Ravi did is to assume that his actions are as rare as Clementi's suicide, that whenever this kind of thing happens, we will hear of it.  This strikes me as burying one's head in the sand.  It's not appealing, because many of us really do want to blame Ravi, but one can't consistently believe both that Ravi inevitably caused Clementi's suicide, and that their situation is common. 

But if the opposite is true, and similar situations are playing themselves out all the time (just with much lighter consequences), then what are we, as a society, to do with Ravi?  What are we to do with anyone who does something criminal that then leads, against (let us say) hundred-to-one odds, to someone's death?  Ostensibly, their actions put them in a lottery of sorts.  We punish the lottery losers, and everyone else goes unscathed, perhaps even unnoticed.

Is this justice?  Does the punishment really fit the crime, or is it more that it fits the consequences?  If it fits the crime, what should we do about those who do not lead to any substantive damage?  On a more abstract level, are we doing what we should to protect potential victims?  Even from the point of view of American jurisprudence, in which the results matter, the situation is unclear.  By throwing the book at Ravi, and missing the others, do we send the message that what Ravi did was wrong?  Or do we just send the message that one just needs to avoid getting caught?

Someday, perhaps, situations like Ravi/Clementi will cease to happen.  It seems unlikely to me, but just perhaps!  But in the meantime, we must think hard about the consequences of punishing people for the results of their crimes, when those results are rare.

No Two Alike

Another meandering post.  You've been warned.

I'm re-reading Isaac Asimov's informal autobiography, I. Asimov (a play on his collection of robot stories, entitled I, Robot, and to be distinguished from his formal autobiographies published earlier in his life), and finding it quite entertaining.  Partly, this is because I'm an inveterate re-reader and re-watcher.  My enjoyment of a piece of writing or a movie or a TV program doesn't diminish because I know how it goes.  If I enjoyed it the first time, I'll enjoy it just as much the seventh time, or the fifty-seventh.  Even a sporting event isn't diminished because I know how the final score (although I do prefer to watch it live the first time, if I can).  All this just by the way.

Anyway, in this book, Asimov mentions his facility at giving impromptu talks, and mentions by way of illustration that he has given a couple of thousand talks, no two exactly alike.

And that phrase, "no two exactly alike," is so characteristic of snowflakes that I immediately thought of them.  In fact, I'd go so far as to wager that if you asked people what the first thing was that they thought about snowflakes, it would be that no two are alike.

But is that actually so?  Have there really never been two snowflakes alike?  If you're like most people, you'd probably just as soon leave well enough alone and assume it's true.  For the heck of it, though, take a trip with me down the rabbit hole.

The whole idea that no two snowflakes are exactly alike has been around for time immemorial, but things really got moving with a man named Wilson Bentley (1865-1931), who grew up in Vermont.  When he was fifteen, his mother gave him an old microscope to experiment with.  Well, Vermont winters being what they were, I suppose it's natural that Wilson should have been drawn to snowflakes.  And so he took to maneuvering snowflakes under his microscope and sketching them.

It turned out, however, that they melted quickly—far too quickly for him to sketch in time.  So Bentley assembled a contrivance, a camera attached to a microscope attached to a board covered with black velvet, which permitted him to take pictures of the snowflakes before they melted.  Over his lifetime, he took images of over five thousand snowflakes, and sure enough, no two of them were exactly alike.

Five thousand, though a lot to take pictures of, is still a minuscule fraction of all the snowflakes that ever were, or even of those that are currently in existence (a constantly changing population, to be sure).  Surely there is no way that we could possibly take pictures of all the ones that currently exist, let alone those that have ever existed.  Is there perhaps another way to answering the question?

Consider: Each year, a substantial portion of the Earth is hit by snowstorms sufficient to dump several meters of snow on the ground.  I'm not sure of my statistics, but we probably wouldn't be far off if we assumed that the total annual snowfall amounted to a depth of, let's say, two tenths of a meter over the entire surface of the Earth, if it was spread around evenly.  Since the surface area of the Earth is about 5×10^14 square meters, we're talking about 10^14 cubic meters of snow.  When packed tightly (tightly enough to crush them), snowflakes might occupy a cube about a tenth of a millimeter on a side.  So each year, we get something like 10^26 snowflakes.  Taking into account the fact that there has been snowfall for a few billion years, there have been perhaps 10^36 snowflakes, ever, in the Earth's history.  That's a lot of snowflakes.

However, there are also lots of different shapes that any particular snowflake might take on.  Snowflakes exhibit six-fold symmetry because they're constructed from ice crystals, which have six-fold symmetry.  (You can find a picture of one in this article.)  So let's represent a snowflake as a hexagonal lattice, a bit like a honeycomb of cells, each of which might be occupied by an ice crystal, or not.  An individual hexagonal ice crystal is a few tenths of a nanometer across, whereas an entire snowflake might be a few tenths of a millimeter across.  So the hexagonal lattice representing our snowflake would have a diameter of about a million cells, and would contain about 750 billion cells in all.

Does that mean that there are nearly a trillion possible snowflakes?  No, because each one of those cells could either have an ice crystal, or not.  We could represent the snowflake by filling each one of those cells with a 1 if it had an ice crystal, or a 0 if it did not.  In other words, each snowflake would be represented by a huge binary number with 750 billion digits.  Such a tremendous number is on the order of 10 raised to the 230 billionth power.

It's hard to overstate how big a number this is.  Even if you were, somehow, to write a 100 digits a second, every second of every hour of every day, without interruption for sleep or eating, you have perhaps only an even-money chance of just writing this number out during your entire lifetime.  It goes without saying that it's much, much, much larger than 10^36.  (It is, however, much smaller than a googleplex.  I just thought I'd point that out.)

However, we're not playing quite fair, because we've completely neglected the symmetry exhibited by most snowflakes.  If we take that into account, it turns out that the number of possible snowflakes drop to something more like 10 raised to the 40 billionth power.  Quite a bit smaller, but still much larger than 10^36.

There's another thing, too.  Bentley took his photographs with an optical microscope, which was of course incapable of resolving ice crystals down to the individual molecular level.  These days, we're now capable of doing that, but it would be unfair to insist that snow crystals, which in an ordinary atmospheric environment would be constantly changing anyway, be identical to that level of precision.  A typical photograph of a snowflake might be able to resolve crystals to a level of detail that would take a hundred thousand cells to fill the entire snowflake.  Remembering to take into account the symmetry of snowflakes, there could still be on the order of 10^5,000 different snowflakes, at this reduced level of resolution.  Still much larger than 10^36.

OK, how about this?  If one looks at an array of Bentley's photographs, one notices that the ice crystals are not arranged haphazardly around the snowflakes, even after one takes into account the six-fold symmetry.  Instead, there is order at all different scales.  In fact, people have likened snowflakes to fractals; there are even simulations of snowflake generation that build upon the fractal arrangement.

That reduces the level of variation accessible to the snowflake.  It's hard to say for sure, but in most of the Bentley images, I think one can make out about six levels of detail.  (That's consistent with a scale ratio of about two to three.)  What's more, each unit of detail has within it detail that only goes about three or four levels down, which means that each level can be represented using about fifty bits or so.  That means a total of three hundred bits might suffice to denote a snowflake to the level of precision needed to figure out whether they match or not.  That would still mean about 10^90 distinct snowflakes, though.

All right, one last thing, which at first will seem to be a significant digression.  There is, in probability, something called the birthday paradox, which goes something like this: Suppose you get fifty otherwise randomly selected people together in a room.  What are the odds that at least one pair of them will share the same birthday (possibly different year)?  One in four?  One in ten?  How many people do you think you need to make the odds even?  Would forty do it?  How about sixty?  A hundred?

The answer, surprising to most people who haven't heard this question before, is that the odds are about even that out of just 23 people, at least one pair will share a birthday in common.  It's a bit surprising because there are 365 days in a year (not counting leap day), but consider what happens if you choose the people one by one.  The first, of course, can have any birthday at all.  In order to avoid a pair sharing the birthday, the second must not share a birthday with the first.  The third must avoid sharing a birthday with both the first and the second.  The fourth must avoid sharing a birthday with the first, the second, and the third.  And so on.  By the time you get to 23 people, there are about 250 birthday sharings that must be independently avoided.  It's not surprising that such sharings are avoided only half the time.

It turns out that this "paradox" (not truly a paradox at all, naturally, but just a counter-intuitive result of probability theory) has very wide applicability.  The number of samples that can be randomly selected before you stand a good chance of getting a pair is much smaller than the total number of choices.  In fact, it's on the same order as the square root of the number of choices.  (There's that square root again!)  The square root of 365 is a bit over 19, and sure enough, 23 isn't very far over 19.  If one takes into account the year of birth over the course of a century, then there are about 36,500 different birthdates, but the square root of 36,500 is only about 191, so that only about 200 randomly selected people are needed before you have a good chance of matching the entire birthdate.  And the square root of 10^90 is 10^45, so the size of the collection of snowflakes you need to have a good chance of pairing two of them is about 10^45.

It's more than 10^36, but not much more.  (What's a factor of a billion between friends?)  And there are a lot of back-of-the-envelope manipulations in what I wrote, so perhaps there are other deeper symmetries to take advantage of.  But I think it's rather magical that the numbers work out nicely so that it's quite possible that somewhere, across the vast mists of time, there were at (probably very different) points, two identical snowflakes!

Slip Sliding Away

Here's a counter-sliding game I came up with a while back, while visiting my parents.

My parents have these small flags of various countries, which can be stood up, UN-style, because they're on flagpoles that are stuck into circular bases.  The flags can be removed from the bases to be waved, and when they are, you're left with just the circular bases.  One day, while idly sliding them around the table, I thought about using them for various geometrical exercises.  Of course, if one doesn't have small circular flagpole bases, one can use any kind of equally sized circular tokens; any kind of circular coin should work just fine.

The rules I set up for myself were as follows:

1. One starts out with two touching counters.  This counts as two moves.  (For "historical reasons.")
2. On any subsequent move, one may add a counter; this counter must touch two existing counters on the table.  (There is an exception, which I will mention later, in connection with an outstanding puzzle.)
3. Or, one may remove a counter.  One must remove the counter by sliding it, however, not by lifting it up off the table.

The following picture shows an example.

 

Here, counters 1 and 2 are placed first.  One may then place counters 3, 4, and 5 in that order.  Removing counters 3 and 4 then leaves a straight line of three counters.  One could not construct that straight line directly, by just putting down counters 1, 2, and 5, because counter 5 would not have been placed in contact with two counters.

One could continue twice more around counter 2, creating a filled hexagon of seven counters.  If, however, one wanted to create a hollow hexagon, one would have to remove that middle counter at some point.  It seems tempting to place one more counter below counters 2 and 5, and then remove counter 2 to place at the last corner of the hexagon, but the following diagram shows why that won't work:

 

The space between the two counters is not wide enough to fit the center counter through (in fact, that space has a width only √3 - 1 = 0.732+ times as wide as necessary), so it cannot be slid out in accordance to Rule 3, above.  You might like to see if you can figure out a solution for the hollow hexagon before reading on.

The trick is to set up support for the fifth corner first, then slide out the center counter to become the fifth corner; the sixth corner is then easily slid into place.  Begin by placing six counters in a parallelogram arrangement:



Then slide counter 2 to touch counters 4 and 6:



Now slide counter 4 into the place previously occupied by counter 2:



Finally, slide counter 1 around to touch counters 2 and 4, at the sixth and last corner of the hexagon.  Voilà!



I leave you with two puzzles, one fairly simple, and one open (that is, unsolved):

1. Follow the above rules to construct a hollow triangle of side 4 (just like the arrangement in ten-pin bowling, but without the center pin), in as few moves as possible.  There is more than one solution.
2. Suppose we add an exception to Rule 2, above: We permit a counter to be placed in an arbitrary location on the table, but with the proviso that no required property of the final arrangement can depend on the exact location of that counter.  (For instance, a construction of a rectangle that depends on a counter being placed somewhere between 1 and 2 counter widths away from another is OK, but one that depends on it being placed exactly 1-1/2 counter widths away is not.)  In that case, is it possible to construct a perfect square of four counters, of any side length?  The sides of the square need not be filled in with any counters.