Who's Counting: A Card Trick and a Religious Hoax

Martin Kruskal, a renowned mathematician and physicist at Rutgers University, died in December 2006. Of his many accomplishments there is an intriguing trick that almost anyone can appreciate.

I explain it here, and, prompted by April Fool's Day, I also sketch a sort of biblical hoax based on it that I first proposed in my 1998 book "Once Upon a Number."

Kruskal's trick can be most easily explained in terms of a well-shuffled deck of cards with all the face cards removed. The deck has 1s (aces) through 10s only. Imagine two players, Hoaxer and Fool. Hoaxer asks Fool to pick a secret number between 1 and 10.

For illustration, let's assume Fool picks 7. Hoaxer goes on to instruct Fool to watch for the card with his secret number -- in this case, the 7th card in the deck -- as Hoaxer slowly turns over the cards one by one.

When the card with the secret number is reached, Hoaxer directs Fool to take the value of this card as his new secret number and to repeat the process.

Thus when the 7th card is reached -- let's assume it's a 5 -- Fool's new secret number becomes 5, and so he watches for the next card corresponding to this new secret number. That is, he watches for the 5th card succeeding it in the deck.

Hoaxer continues to slowly turn over the cards one by one. When the 5th succeeding card is reached -- let's assume it's a 9 -- Fool's new secret number becomes 9, and so he watches for the card corresponding to this new secret number. That is, he watches for the 9th card succeeding it in the deck, and so on and on.

As they near the end of the deck, Hoaxer turns over a card and announces, "This is your new secret number," and he is almost always correct.

Why Does It Work?

The deck is not marked or ordered in any way, there are no confederates, there is no sleight of hand, and there is no careful observation of Fool's reactions as he watches the cards being turned over.

How does Hoaxer accomplish this feat? The answer is cute.

At the beginning of the trick, Hoaxer picks his own secret number. He then follows the same instructions he's given to Fool. If he picked a 3 as his secret number, he watches for the 3rd card and notes its value -- say it's a 6 -- which becomes his new secret number. He then looks for the 6th card after it -- say it's a 4 -- and that becomes his new secret number, and so on and on.

Even though there is only one chance in 10 that Hoaxer's original secret number is the same as Fool's original secret number, it is reasonable to assume, and it can be proved, that sooner or later their secret numbers will coincide. That is, 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 Fool and Hoaxer are using the same rule to generate new secret numbers from old. Thus all Hoaxer 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 Fool's secret number as well.

Note that the trick works just as well if there is more than one Fool or even if there is no Hoaxer at all (as long as the cards are turned over one by one by someone).

If there is 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 the numbers will move in lockstep.

My Proposal for a Religious Hoax

Consider now 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; keep on doing this until the climactic and especially sacred word -- say "God" or "heaven" -- is reached. (The letter count of each word plays the role of the numbered cards.)

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 among some that divine inspiration is the only explanation for the fact that the procedure always ends on the sacred word. If the generating rule were more complicated than the simple one above, the effect would be even more mysterious.

The reader can experiment with his own examples or check out the August 1998, issue of Scientific American, where puzzlemeister Martin Gardner, who graciously blurbed my book, came up with an elegant illustration of its proposal for a religious hoax.

Martin Kruskal, I should note, was innocent of perpetrating any hoaxes. He was simply a very fine applied mathematician.

In any case, Happy April Fool's Day to Hoaxers and Fools alike.

John Allen Paulos, a professor of mathematics at Temple University, has written such best-sellers as "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.