## Brothers in Binary

A number is said to be *perfect* if it equals the sum of its divisors: 6 is divisible by 1, 2, and 3, and 1 + 2 + 3 = 6.

St. Augustine wrote, “Six is a number perfect in itself, and not because God created all things in six days; rather the converse is true; God created all things in six days because this number is perfect, and it would have been perfect even if the work of the six days did not exist.”

Perfect numbers are rare. No one knows whether an infinite quantity exist, and no one knows whether any of them are odd. The early Greeks knew the first four, and in the ensuing two millennia we’ve uncovered only 44 more. But they have one thing in common — they reveal a curious harmony when expressed in base 2:

## Unquote

“We have not the reverent feeling for the rainbow that a savage has, because we know how it is made. We have lost as much as we gained by prying into that matter.” — Mark Twain

“At last I fell fast asleep on the grass & awoke with a chorus of birds singing around me, & squirrels running up the trees & some Woodpeckers laughing, & it was as pleasant a rural scene as ever I saw, & I did not care one penny how any of the beasts or birds had been formed.” — Charles Darwin, letter to his wife, April 28, 1858

## Greetings

Launched in 1976, NASA’s Laser Geodynamic Satellite needed a stable orbit to permit precise measurements of continental drift, so its designers gave it a high trajectory and a heart of solid brass. As a result, it’s not expected to return to Earth for 8 million years. That raised an interesting challenge: What message could we attach to the satellite that might be intelligible to our descendants or successors, who might recover it thousands of millennia in the future?

Tasked with that problem, Carl Sagan came up with the “greeting card” at left, which is affixed to LAGEOS on a small metal plaque. Using it, whoever comes upon the plaque can calculate roughly the time between his own epoch and ours. In Sagan’s words, the card says, “A few hundred million years ago the continents were all together, as in the top drawing. At the time LAGEOS was launched the map of the Earth looks as in the middle drawing. Eight million years from now, when LAGEOS should return to Earth, we figure the continents will appear as in the bottom drawing. Yours truly.”

## Balance

For any rectangle, the sum of the squares of the distances from any point P to two opposite corners is equal to the sum of the squares of the distances from that point to the two other corners (so, above, *a*^{2} + *c*^{2} = *b*^{2} + *d*^{2}). This remains true whether the point is inside or outside the rectangle, on a side or a corner, or even outside the plane.

Pushkin wrote, “Inspiration is needed in geometry, just as much as in poetry.”

## Math Notes

10989 × 9 = 98901 × 1

21978 × 8 = 87912 × 2

32967 × 7 = 76923 × 3

43956 × 6 = 65934 × 4

54945 × 5 = 54945 × 5

## Correlation, Causation

From Richard F. Mould’s *Introductory Medical Statistics* — this graph plots the population of Oldenburg, Germany, at the end of each year 1930-1936 against the number of storks observed in that year.

Does this explain the storks’ presence? Not necessarily: In 1888 J.J. Sprenger noted, “In Oldenburg there is a curious theory that the autumnal gatherings of the storks are in reality Freemasons’ meetings.”

## Digit Work

A useful system of finger reckoning from the Middle Ages:

To multiply 6 x 9, hold up one finger, to represent the difference between the 5 fingers on that hand and the first number to be multiplied, 6.

On the other hand, hold up four fingers, the difference between 5 and 9.

Now add the number of extended fingers on each hand to get the first digit of the answer (1 + 4 = 5), and multiply the number of closed fingers on each hand to get the second (4 × 1 = 4). This gives the answer, 54.

In this way one can multiply numbers between 6 and 9 while knowing the multiplication table only up to 5 × 5.

A similar system could be used to multiply numbers between 10 and 15. To multiply 14 by 12, extend 4 fingers on one hand and 2 on the other. Add them to get 6; add 10 times that sum to 100, giving 160; and then add the product of the extended fingers, 4 × 2, to get 168.

This system reflects the fact that *xy* = 10 [(*x* – 10) + (*y* – 10)] + 100 + (*x* – 10)(*y* – 10).

(From J.T. Rogers, *The Story of Mathematics*, 1968.)

## Misc

- The first child to be vaccinated in Russia was named Vaccinov.
- Every treasurer of the United States since 1949 has been a woman.
- 15642 = 1 + 5
^{6}+ 4^{2} - up inverted is dn.
- “Life well spent is long.” — Leonardo

## Stormy Weather

Take an ordinary magic square and imagine that the number in each cell denotes its altitude above some common underlying plane. And now suppose that it begins to rain, with an equal amount of water falling onto each cell. What happens? In the square at left, the water cascades from square 25 down to square 21, and thence down to 10, 7, 2, and into space; because there are no “lowlands” on this landscape, no water is retained. (Water flows orthogonally, not diagonally, and it pours freely over the edges of the square.)

By contrast, in the square on the right a “pond” forms that contains 69 cubic units of water — as it happens, the largest possible pond on a 5×5 square.

With the aid of computers, these imaginary landscapes can be “terraformed” into surprisingly detailed shapes. Craig Knecht, who proposed this area of study in 2007, created this 25×25 square in 2012:

Next year will mark the 500th anniversary of the famously fertile magic square in Albrecht Dürer’s 1514 engraving *Melancholia* — a fact that Knecht has commemorated in the shape of the ponds on the 14×14 square at right.

## Partial Credit

On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”

At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.

When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.”

## Finding Yourself

Twinkle, twinkle, little star,

How I wonder what you are.

Up above the world so high,

Like a diamond in the sky.

Twinkle, twinkle, little star,

How I wonder what you are.

Choose any word in the first two lines, count its letters, and count forward that number of words. For example, if you choose STAR, which has four letters, you’d count ahead four words, beginning with HOW, to reach WHAT. Count the number of letters in that word and count ahead as before. Continue until you can’t go any further. You’ll always land on YOU in the last line.

See Finding Religion and The Kruskal Count.

## Soulmates

In 1966, asked to describe the person least likely to develop atherosclerosis, Cambridge research fellow Alan N. Howard answered, “A hypotensive, bicycling, unemployed, hypo-beta-lipoproteinic, hyper-alpha-lipoproteinic, non-smoking, hypolipaemic, underweight, premenopausal female dwarf living in a crowded room on the island of Crete before 1925 and subsisting on a diet of uncoated cereals, safflower oil, and water.”

Oxford physician Alan Norton added that her male counterpart was an ectomorphic Bantu who worked as a London bus conductor, had spent the war in a Norwegian prison camp, never ate refined sugar, never drank coffee, always ate five or more small meals a day, and was taking large doses of estrogen to check the growth of his prostate cancer.

“All these phrases mark correlations established in the last few years in a field of medical research which, in volume at least, is unsurpassed,” noted Richard Mould in *Mould’s Medical Anecdotes*. “The conflict of evidence is unequalled as well.”

## Ho

*If this sentence is true, then Santa Claus exists.*

If that sentence is true, then it’s the case that Santa Claus exists. But wait — in making this observation, we seem to have confirmed the truth of the original sentence. And if that sentence is true, then Santa Claus exists! Where is the error?

(By Raymond Smullyan.)

## Head or Tail

Cihan Altay’s Rotator typeface presents the digits 0-9 whether it’s right side up or upside down.

So this equation:

… can be inverted to make this one:

Both are valid.

(Thanks, Lorenzo.)

## Round Numbers

Every now and again one comes across an astounding result that closely relates two foreign objects which seem to have nothing in common. Who would suspect, for example, that on the average, the number of ways of expressing a positive integer

nas a sum of two integral squares,x^{2}+y^{2}=n, is π?

— Ross Honsberger, *Mathematical Gems III*, 1977

## Times Square

Prove that the product of four consecutive positive integers cannot be a perfect square.

## Dinner Charge

A curious effect produced by lightning is described to us by Dr. Enfield, writing from Jefferson, Iowa, U.S. A house which he visited was struck by lightning so that much damage was done. After the occurrence, a pile of dinner plates, twelve in number, was found to have every other plate broken. It would seem as if the plates constituted a condenser under the intensely electrified condition of the atmosphere. The particulars are, however, so meagre that it is difficult to decide whether the phenomenon was electrical or merely mechanical.

— *Nature*, June 12, 1902

## A Body at Rest

In an anonymous letter to the London *Times* in 1825, Thomas Steele of Magdalen College, Cambridge, proposed enshrining Isaac Newton’s residence in a stepped stone pyramid surmounted by a vast stone globe. The physicist himself had died more than a century earlier, in 1727, and lay in Westminster Abbey, but Steele felt that preserving his home would produce a monument “not unworthy of the nation and of his memory”:

When travelling through Italy, I was powerfully struck by the unique situation and singular appearance of the Primitive Chapel at

Assisi, founded by St. Francis.As you enter the porch of the great Franciscan church, you view before you this small cottage-like chapel, standing directly under the dome, and perfectly isolated.

Now, Sir, among the many splendid improvements which are making in the capital, would it not be a noble, and perhaps the most appropriate, national monument which could be erected, if an azure hemispherical dome, or what would be better, a portion of a sphere greater than a hemisphere, supported on a massive base, were to be reared, like that of

Assisi, over thehouse and observatoryof the writer of thePrincipia?The house might be fitted up in such a manner as to contain a council-chamber and library for the Royal Society; and it is perhaps not unworthy of being remarked, that it is not more than about two hundred yards distant from the University Club House.

Protected, by the means which I have described, from the dilapidating influence of rains and winds, the venerable edifice in which Newton studied, or was inspired, — that ‘palace of the soul,’ might stand fast for ages, a British monument more sublime than the Pyramids, though remote antiquity and vastness be combined to create their interest.

Steele wasn’t an architect, and he left the details to others, but he was imagining something enormous: In a subsequent letter to the *Mechanics’ Magazine* he wrote that “the base of my design [appears] to coincide with the base of St. Paul’s (a sort of crude coincidence of course, in consequence of the angle at the transept), and that the highest point of my designed building should, at the same time, appear to coincide with a point of the great tower of the cathedral, about 200 feet high — the height of the building which I propose to have erected.”

The plan never went forward, but the magazine endorsed the idea: “We need scarcely add, that there is no description of embellishment which might not be with ease introduced into the structure, so as to render it as perpetual a monument to the taste as it would be to the national spirit and gratitude of the British people.”

## Warm Words

Future president Herbert Hoover published a surprising title in 1912: An English translation of the 16th-century mining textbook *De Re Metallica*, composed originally by Georg Bauer in 1556. Bauer’s book had remained a classic work in the field for two centuries, with some copies deemed so valuable that they were chained to church altars, but no one had translated the Latin into good modern English. Biographer David Burner wrote, “Hoover and his wife had the distinct advantage of combining linguistic ability with mineralogical knowledge.”

Hoover, a mining engineer, and his wife Lou, a linguist, spent five years on the project, visiting the areas in Saxony that Bauer had described, ordering translations of related mining books, and spending more than $20,000 for experimental help in investigating the chemical processes that the book described.

The Hoovers offered the 637-page work, complete with the original woodcuts, to “strengthen the traditions of one of the most important and least recognized of the world’s professions.” Of the 3,000 copies that were printed, Hoover gave away more than half to mining engineers and students.

## Yablo’s Paradox

All the statements below this one are false.

All the statements below this one are false.

All the statements below this one are false.

All the statements below this one are false.

All the statements below this one are false.

…

These statements can’t all be false, because that would make the first one true, a contradiction. But neither can any one of them be true, as a true statement would have to be followed by an infinity of false statements, and the falsity of any one of them implies the truth of some that follow. Thus there’s no consistent way to assign truth values to all the statements.

This is reminiscent of the well-known liar paradox (“This sentence is false”), except that none of the sentences above refers to itself. MIT philosopher Stephen Yablo uses it to show that circularity is not necessary to produce a paradox.

## Finding Mates

A remarkable spelling trick by American magician Howard Adams:

From a deck of cards choose five cards and their mates. A card’s mate is the card of the same value and color; for example, the mate of the five of clubs is the five of spades.

Arrange the cards in the order ABCDEabcde, where ABCDE are the chosen cards and abcde are the mates. Cut this packet as many times as you like, then deal five cards onto the table, reversing their order. Place the remaining five cards beside them in a second pile.

Now spell the phrase LAST TWO CARDS MATCH. As you say “L,” choose either pile at random and transfer a card from the top to the bottom. Do the same for A, S, and T. Now remove the top card from each pile and set them aside as a pair.

Perform the same procedure as you spell TWO, CARDS, and MATCH. When you’re finished, two cards will remain on the table. Not only do these cards match, but so do each of the other pairs!

## Well Done

Using a 7-quart and a 3-quart jug, how can you obtain exactly 5 quarts of water from a well?

That’s a water-fetching puzzle, a familiar task in puzzle books. Most such problems can be solved fairly easily using intuition or trial and error, but in *Scripta Mathematica*, March 1948, H.D. Grossman describes an ingenious way to generate a solution geometrically.

Let *a* and *b* be the sizes of the jugs, in quarts, and *c* be the number of quarts that we’re seeking. Here, *a* = 7, *b* = 3, and *c* = 5. (*a* and *b* must be positive integers, relatively prime, where *a* is greater than *b* and their sum is greater than *c*; otherwise the problem is unsolvable, trivial, or can be reduced to smaller integers.)

Using a field of lattice points (or an actual pegboard), let O be the point (0, 0) and P be the point (*b*, *a*) (here, 3, 7). Connect these with *OP*. Then draw a zigzag line *Z* to the right of *OP*, connecting lattice points and staying as close as possible to *OP*. Now “It may be proved that the horizontal distances from *OP* to the lattice-points on *Z* (except *O* and *P*) are in some order without repetition 1, 2, 3, …, *a* + *b* – 1, if we count each horizontal lattice-unit as the distance *a*.” In this example, if we take the distance between any two neighboring lattice points as 7, then each of the points on the zigzag line *Z* will be some unique integer distance horizontally from the diagonal line *OP*. Find the one whose distance is *c* (here, 5), the number of quarts that we want to retrieve.

Now we have a map showing how to conduct our pourings. Starting from *O* and following the zigzag line to *C*:

- Each horizontal unit means “Pour the contents of the
*a*-quart jug, if any, into the*b*-quart jug; then fill the*a*-quart jug from the well.” - Each vertical unit means “Fill the
*b*-quart jug from the*a*-quart jug; then empty the*b*-quart jug.”

So, in our example, the map instructs us to:

- Fill the 7-quart jug.
- Fill the 3-quart jug twice from the 7-quart jug, each time emptying its contents into the well. This leaves 1 quart in the 7-quart jug.
- Pour this 1 quart into the 3-quart jug and fill the 7-quart jug again from the well.
- Fill the remainder of the 3-quart jug (2 quarts) from the 7-quart jug and empty the 3-quart jug. This leaves 5 quarts in the 7-quart jug, which was our goal.

You can find an alternate solution by drawing a second zigzag line to the left of *OP*. In reading this solution, we swap the roles of *a* and *b* given above, so the map tells us to fill the 3-quart jug three times successively and empty it each time into the 7-quart jug (leaving 2 quarts in the 3-quart jug the final time), then empty the 7-quart jug, transfer the remaining 2 quarts to it, and add a final 3 quarts. “There are always exactly two solutions which are in a sense complementary to each other.”

Grossman gives a rigorous algebraic solution in “A Generalization of the Water-Fetching Puzzle,” *American Mathematical Monthly* 47:6 (June-July 1940), pp. 374-375.

## A Magic Mystery

In 1991, David Collison sent this figure to Canadian magic-square expert John Henricks, with no explanation, and then died.

It’s believed to be the first odd-ordered bimagic square ever discovered. Each row, column, and diagonal produces a sum of 369. The square remains magic if each number is squared, with a magic sum of 20,049.

No one knows how Collison created it.

UPDATE: Wait, Collison’s wasn’t the first — G. Pfeffermann published a 9th-order bimagic square as a puzzle in *Les Tablettes du Chercheur* in 1891. (Thanks, Baz.)

## Regrets

On June 28, 2009, Stephen Hawking hosted a party for time travelers, but he sent out the invitations only afterward.

No one turned up.

He offered this as experimental evidence that time travel is not possible.

“I sat there a long time,” he said, “but no one came.”