Some coincidences are significant but not for the ostensible reasons given. There is, for example, a magic trick of recent vintage that nicely illustrates a way in which two people can "cognitively couple" and generate an otherwise hard to explain coincidental merging. As explained in the last paragraph below, it also leads to a proposal for a new biblical hoax.

The card trick was invented about 15 years ago by the physicist Martin Kruskal and can be most easily explained in terms of a deck of cards with all the face cards removed. Imagine two players, Tricked and Trickster. Trickster asks Tricked to pick a secret number - say it's X - between 1 and 10 and goes on to instruct Tricked to watch for the Xth card as Trickster slowly and one by one turns over the cards in a well-shuffled deck. When the Xth card is reached - say it's a Y - it becomes Tricked's new secret number and he is asked to watch for the Yth succeeding card after it, as Trickster continues to slowly turn over the cards one by one. When the Yth succeeding card turns up, its value - say it's Z - becomes Tricked's new secret number, and again he is asked to watch for the Zth card succeeding after it for his new secret number, and so on.

Thus if Tricked first picks 7 as his secret number, he would watch for the 7th card as Trickster slowly turns the cards over. If the 7th card is a 5, his new secret number would become a 5, and he would watch for the 5th card after it. If the 5th card after this is a 10, 10 would become his new secret number and he would watch for the 10th card after it to determine his new secret number. As they near the end of the deck, Trickster turns over a card and announces, "This is your present secret number," and he is almost always correct.

The deck is not marked or ordered, there are no confederates, there is no sleight of hand, and there is no careful observation of Tricked's reactions as he watches the cards being turned over. How does Trickster accomplish this feat? The answer is cute. At the beginning of the trick, Trickster picks his own secret number. He then follows the same instructions he's given to Tricked. If he picked a 3 as his secret number, he watches for the 3rd card and notes its value - say it's a 9 - which becomes his new secret number. He then looks for the 9th card after it - say it's a 4 - and that becomes his new secret number.

Even though there is only one chance in 10 that Trickster's original secret number is the same as Tricked's original secret number, it is reasonable to assume and can be proved that sooner or later their secret numbers will coincide. I.e., if two more or less random sequences of secret numbers between 1 and 10 are selected, sooner or later they will, simply by chance, lead to the same card. Furthermore, from that point on, the secret numbers will be identical since both Tricked and Trickster are using the same rule to generate new secret numbers from old. Thus all Trickster does is wait until he nears the end of the deck and then turn over the card corresponding to his last secret number, confident that by that point it will probably be Tricked's secret number as well.

Aside from the pleasures of understanding it, does this trick have any real-world analogues? Note that the trick works just as well if there is more than one Tricked person or even if there is no Trickster at all (as long as the cards are turned over one by one by someone). If there are a large number of people and each picks his or her own initial secret number and generates a new one from the old one in accordance with the procedure above, all of them will eventually have the same secret number and thereafter will move in lockstep.

If we now allow people's new secret number to be determined in a more complicated way from several of its predecessor secret numbers instead of just from its immediate predecessor, and if we change the scenario from turning cards over one by one to some other sequential and numerical activity such as the lottery or the stock market, we see the potential for lockstep behavior on a large scale to develop naturally. If, for example, there are many investors who use the same computer software, i.e., the same rules for determining when to buy or sell, it is conceivable that some attenuated variant of the above might result whatever the investors' initial positions.

I THUS PROPOSE THE FOLLOWING RELIGIOUS HOAX. Consider a holy book with the compelling property that no matter what word from the early part of the book is chosen, the following procedure always leads to the same climactic and especially sacred word: Begin with whatever word you like; count the letters in it; say this number is X; proceed forward X words to another word; count the letters in it; say this number is Y; proceed forward Y words to another word; count the letters in it; say this number is Z; iterate this procedure until the climactic and especially sacred word is reached. It's not too hard to imagine frenzied checking of this procedure using word after word from the early part of the holy book and the increasing certainty that divine inspiration is the only explanation for the phenomenon. If the generating rule were more complicated than the simple one above, the effect would be even more mysterious.

Martin Gardner was gracious enough to provide a blurb for Once Upon a Number and was inspired by the excerpt above to come up with an elegant illustration of the recipe in the August, 1998 issue of Scientific American.

Also in Once Upon a Number is a discussion of Michael Drosnin's book, The Bible Code, whose claims are clarified by a discussion of a lesser-known recent discovery that has been suppressed by President Clinton's lawyers. Imbedded within the U.S. Constitution is a so-called equidistant letter sequence foretelling the Lewinsky sex scandals!

Back to Paulos Home Page