Science & Math

Infinite Composition

In Tristram Shandy, the title character laments that he’ll never be able to finish his autobiography, as he seems to need a year to record each day’s events. “It must follow, an’ please your worships, that the more I write, the more I shall have to write.”

But Bertrand Russell noted that if Shandy’s eventful life had lasted forever, no part of his biography would have remained unwritten — for the hundredth day would be recorded in the hundredth year, the thousandth in the thousandth, and so on. “This paradoxical but perfectly true proposition depends upon the fact that the number of days in all time is no greater than the number of years.”


Rest the ends of a yardstick on your index fingers. Now slowly draw your fingers together, trying to make them meet at some spot other than the center of the stick.

It’s impossible. When either finger leads, it bears more weight, which creates more friction, and the other catches up.

Math Notes

651 × 156 = 372 × 273

The Kruskal Count

Here’s a card trick devised by Rutgers physicist Martin Kruskal. Give a friend a deck of cards and ask her to follow these instructions:

  1. Think of a “secret number” from 1 to 10. (Example: 6)
  2. Shuffle the deck and deal the cards face up one at a time, counting silently as you go.
  3. When you reach the secret number, note the value of that card and adopt it as your new secret number. Aces count as 1; face cards count as 5. (Example: If the 6th card is a 4, then 4 becomes your new secret number.)
  4. Continue dealing, counting silently anew from 1 each time you adopt a new number. Remember the last secret card you reach.

That’s it. You just stand there and watch her deal. When she’s finished, you can identify her final secret card in any way you please, preferably through a grotesquely extortionate wager.

You can do this because you’ve simply played along. When she’s dealing, note the value of an early card and then silently follow the same steps that she is. Five times out of six, your “paths” through the deck will intersect and your final secret card will match hers. That’s far from obvious, though; the trick can be baffling if you refuse to explain it.

Kaprekar’s Constant

Choose four distinct digits and arrange them into the largest and smallest numbers possible (e.g., 9751 and 1579). Subtract the smaller from the larger to produce a new number (9751 – 1579 = 8172) and repeat the operation.

Within seven iterations you’ll always arrive at 6174.

With three-digit numbers you’ll aways arrive at 495.

Math Notes

410 + 610 + 710 + 910 + 310 + 010 + 710 + 710 + 710 + 410 = 4679307774


Abraham de Moivre correctly predicted the date of his own death.

He noted that he was sleeping 15 minutes longer each day and surmised that he would die on the day he slept for 24 hours. That date, he calculated, would be Nov. 27, 1754.

He was right.

Food for Thought

A force of 1 newton is about the weight of an apple.

Burglars Beware

Who says math is too abstract?

The Chvátal Art Gallery Theorem states that if you run an art gallery with n corners, you’ll need n/3 guards (at most) to watch the entire gallery—regardless of its shape.

Math Notes

28 + 48 + 68 + 78 + 88 + 08 + 58 + 08 = 24678050


Pick a three-digit number (example: 412).

Double it to create a six-digit number (412412).

Divide the result successively by 7, by 11, and by 13. There will be no remainders.

The result is the original number.

Where’s the Fallacy?

square-mile fallacy

“In Event of Moon Disaster”

On July 18, 1969, two days before the first lunar landing, presidential speechwriter William Safire composed the following text to be read by President Nixon if astronauts Neil Armstrong and Edwin Aldrin were stranded on the moon:

Fate has ordained that the men who went to the moon to explore in peace will stay on the moon to rest in peace.

These brave men, Neil Armstrong and Edwin Aldrin, know that there is no hope for their recovery. But they also know that there is hope for mankind in their sacrifice.

These two men are laying down their lives in mankind’s most noble goal: the search for truth and understanding.

They will be mourned by their families and friends; they will be mourned by the nation; they will be mourned by the people of the world; they will be mourned by a Mother Earth that dared send two of her sons into the unknown.

In their exploration, they stirred the people of the world to feel as one; in their sacrifice, they bind more tightly the brotherhood of man.

In ancient days, men looked at the stars and saw their heroes in the constellations. In modern times, we do much the same, but our heroes are epic men of flesh and blood.

Others will follow, and surely find their way home. Man’s search will not be denied. But these men were the first, and they will remain the foremost in our hearts.

For every human being who looks up at the moon in the nights to come will know that there is some corner of another world that is forever mankind.

Safire also suggested that Nixon call the “widows-to-be” before the speech, and that a clergyman should commend the astronauts’ souls to the “deepest of the deep” when communications ended.

Math Notes

17 + 77 + 47 + 17 + 77 + 27 + 57 = 1741725

Subtraction Pangram

Each term in this equation contains each of the nine digits once:

subtraction pangram

Proof That 2 Does Not Exist

2 is the only even prime.

But the total number of primes is infinite.

Therefore the probability that a given prime number is even is 1 over infinity, or zero.

Hence it’s impossible for a prime number to be even — and 2 does not exist.

Math Notes

55 + 45 + 75 + 45 + 85 = 54748

The Barber’s Dictum

Let’s say that the densest human head of hair contains 200,000 strands, and that the human population is 6 billion. That means there’s a group of at least 30,000 people today who have precisely the same number of hairs on their heads.

Do you see why?

Where’s the Fallacy?



Suppose the earth were a perfect sphere and you fitted a belt around its equator.

The belt would be 40 million meters long. If you now increased its length by a mere 5 meters, how high would it ride above the earth’s surface?

The answer, surprisingly, is 0.8 meters — well above the current limbo record.

Square Deal

Look! Canceling exponents works even in algebra:

canceling exponents in algebra

Math Notes

1634 = 14 + 64 + 34 + 44

The Declining Dollar

declining dollar

Numerical Pangrams

A pangram is a sentence that uses each letter of the alphabet exactly once:


“Carved symbols in a mountain hollow and on the bank of a fjord irritated an eccentric person.” They’re a bit awkward in English, so here’s the same idea using numbers. Each of these (valid) equations uses the digits 1-9 exactly once:

42 × 138 = 5796
27 × 198 = 5346
39 × 186 = 7254
48 × 159 = 7632
28 × 157 = 4396
4 × 1738 = 6952
4 × 1963 = 7852

Even better: The numbers 3 and 51249876, between them, use all 9 digits — and so does their product, 153749628.