# More Fortuitous Numbers

Two years ago I wrote about the number 84,672, which has a surprising property: When its name is written out in (American) English (EIGHTY FOUR THOUSAND SIX HUNDRED SEVENTY TWO) and the letter counts of those words are multiplied together (6 × 4 × 8 × 3 × 7 × 7 × 3), they yield the original number (84,672).

Such numbers are called fortuitous, and, not surprisingly, very few of them are known. When I wrote about them in 2019, the whole list ran 4, 24, 84672, 1852200, 829785600, 20910597120, 92215733299200. Now Jonathan Pappas has discovered two more:

1,239,789,303,244,800,000

ONE QUINTILLION TWO HUNDRED THIRTY NINE QUADRILLION SEVEN HUNDRED EIGHTY NINE TRILLION THREE HUNDRED THREE BILLION TWO HUNDRED FORTY FOUR MILLION EIGHT HUNDRED THOUSAND

3 × 11 × 3 × 7 × 6 × 4 × 11 × 5 × 7 × 6 × 4 × 8 × 5 × 7 × 5 × 7 × 3 × 7 × 5 × 4 × 7 × 5 × 7 × 8 = 1,239,789,303,244,800,000

887,165,996,513,213,819,259,682,435,576,627,200,000,000

EIGHT HUNDRED EIGHTY SEVEN DUODECILLION ONE HUNDRED SIXTY FIVE UNDECILLION NINE HUNDRED NINETY SIX DECILLION FIVE HUNDRED THIRTEEN NONILLION TWO HUNDRED THIRTEEN OCTILLION EIGHT HUNDRED NINETEEN SEPTILLION TWO HUNDRED FIFTY NINE SEXTILLION SIX HUNDRED EIGHTY TWO QUINTILLION FOUR HUNDRED THIRTY FIVE QUADRILLION FIVE HUNDRED SEVENTY SIX TRILLION SIX HUNDRED TWENTY SEVEN BILLION TWO HUNDRED MILLION

5 × 7 × 6 × 5 × 12 × 3 × 7 × 5 × 4 × 11 × 4 × 7 × 6 × 3 × 9 × 4 × 7 × 8 × 9 × 3 × 7 × 8 × 9 × 5 × 7 × 8 × 10 × 3 × 7 × 5 × 4 × 10 × 3 × 7 × 6 × 3 × 11 × 4 × 7 × 6 × 4 × 11 4 × 7 × 7 × 3 × 8 × 3 × 7 × 6 × 5 × 7 × 3 × 7 × 7 = 887,165,996,513,213,819,259,682,435,576,627,200,000,000

A 10th solution, if one exists, will be greater than 10138.

Details are here. Jonathan has also discovered some cyclic solutions (ONE HUNDRED SIXTY EIGHT -> FIVE HUNDRED TWENTY FIVE -> SIX HUNDRED SEVENTY TWO -> FOUR HUNDRED FORTY ONE -> FOUR HUNDRED TWENTY -> ONE HUNDRED SIXTY EIGHT) and the remarkable 195954154450774917120 -> 195954154450774917120000 -> 1959541544507749171200000.

(Thanks, Jonathan.)

# On and Off

Wyoming’s Intermittent Spring is well named: It runs and stops alternately, in segments of about 15 minutes. The mechanism isn’t known for certain, but scientists suspect that a cavern in the rock face is filling continuously with groundwater, and when the water reaches a certain height it spills through a tube that empties into the valley. This produces a natural siphon effect that draws down the reservoir until air enters the tube, which disables the siphon until the cycle starts again. University of Utah hydrologist Kip Solomon says, “We can’t think of another explanation at the moment.”

Solomon’s tests show that the spring water has been exposed to air underground, which lends support to the theory.

# Direction

Ancient Egypt was an essentially one-dimensional country strung out along the Nile, which flows from south to north. The winds were conveniently arranged to be predominantly northerly. To go north, a traveler could let his boat drift, while with a sail he could move south against the slow current. For this reason, in the writing of the ancient Egyptians, ‘go downstream (north)’ was represented by a boat without sails, and ‘go upstream (south)’ by a boat with sails. The words (and concepts) or north-south and up-downstream became merged. Since the Nile and its tributaries were the only rivers known to the ancient Egyptians, this caused no difficulties until they reached the Euphrates, which happened to flow from north to south. The resulting confusion in the ancient Egyptian mind is recorded for us to read today in their reference to ‘that inverted water which goes downstream (north) in going upstream (south).’

— P.L. Csonka, “Advanced Effects in Particle Physics,” Physical Review, April 1969, 1266-1281

# Eye Music

The scores of George Crumb’s Makrokosmos piano suite take unusual forms: a circle, a cross, a spiral, a peace sign.

Each is handwritten.

# Oddity

Discovered by Galileo:

$\displaystyle \frac{1}{3} = \frac{1 + 3}{5 + 7} = \frac{1 + 3 + 5}{7 + 9 + 11} = \cdots .$

A “proof without words” is here.

# A Self-Characterizing Figure

This is pretty: If you choose n > 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals n.

(Andre P. Mazzoleni and Samuel Shan-Pu Shen, “The Product of Chord Lengths of a Circle,” Mathematics Magazine 68:1 [February 1995], 59-60.)

(Demonstration by Jay Warendorff.)

# Meditation

Written by Albert Einstein at the invitation of a German magazine, 1921:

What Artistic and Scientific Experience Have in Common

Where the world ceases to be the scene of our personal hopes and wishes, where we face it as free beings admiring, asking, and observing, there we enter the realm of Art and Science. If what is seen and experienced is portrayed in the language of logic, we are engaged in science. If it is communicated through forms whose connections are not accessible to the conscious mind but are recognized intuitively as meaningful, then we are engaged in art. Common to both is the loving devotion to that which transcends personal concerns and volition.

(From Helen Dukas and Banesh Hoffmann, eds., Albert Einstein, the Human Side: New Glimpses From His Archives, 1979.)

# Bright Idea

A jailer will send each of a group of n prisoners alone into a certain room. Each prisoner will visit the room infinitely often, but the order of the visits will be determined arbitrarily by the jailer. The prisoners can confer in advance, but once the visits have commenced they can communicate with one another only by means of a light in the room, which they can turn on or off. How can they ensure that some prisoner will eventually be able to determine that everyone has visited the room?

# The Endless House

In 1924 architect Frederick Kiesler proposed a house fashioned as a continuous shell rather than an assembly of columns and beams:

The house is not a dome, but an enclosure which rises along the floor uninterrupted into walls and ceilings. Thus the safety of the house does not depend on underground footings but on a strict coordination of all enclosures of the house into one structural unit. Even the window areas are not standardized in size or shape, but are, rather, large and varied in their transparency and translucency, and form part of the continuous flow of the shell.

Storage space is provided between double shells in the walls, radiant heating is built into the floor, and there are no separate bathrooms, as bathing is done in the individual living quarters.

“While the concept of the house does not advocate a ‘return to nature’, it certainly does encourage a more natural way of living, and a greater independence from our constantly increasing automative way of life.”

(Via Ulrich Conrads, The Architecture of Fantasy: Utopian Building and Planning in Modern Times, 1962.)

# Certainty

“Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe that that bridge would fall down?” — Frank Ramsey, to Wittgenstein

“Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether, say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.” — Richard W. Hamming