## Math Notes

By Royal V. Heath:

12 + 43 + 65 + 78 = 87 + 56 + 34 + 21

That’s not terrifically impressive on its face. But:

- Each side uses the digits 1-8.
- The whole equation is a palindrome.
- It remains valid if you square each term.

## Teamwork

At the start of her career, NIH immunologist Polly Matzinger disliked writing in the passive voice and felt too insecure to adopt the first person. So she listed her dog, Galadriel Mirkwood, as a coauthor and wrote as “we.”

Their paper was published in 1978 in the *Journal of Experimental Medicine*. When the editor learned Galadriel’s species, he barred Matzinger from his pages for the rest of his life.

## Sibling Rivalry

Do men have more sisters than women do? Intuitively it seems they must. In a family with two children, a boy and a girl, the boy has a sister but the girl doesn’t. In a family with four children, two boys and two girls, each boy has two sisters but each girl has one. It seems inevitable that, on average, men must have more sisters than women.

But it isn’t true. There are four possible two-child families, all equally likely: BB, BG, GB, GG. Half of the children in these families have a sibling of the same sex, and half have a sibling of the opposite sex. This observation can be extended to larger families. So men have the same number of sisters as women.

Cut the Knot has a good discussion of the statistics, including a javascript simulator.

## A Population Puzzle

If I have two children, there’s a 50 percent chance that I’ll have a boy and a girl.

But if I have four children, the chance that I’ll have an equal number of boys and girls drops to 6 in 16, or 37.5 percent.

This trend continues — as the number of offspring rises, the chance of having precisely the same number of boys and girls drops:

2 children: 50 percent

4 children: 37.5 percent

8 children: 27.34375 percent

16 children: 19.6380615 percent

32 children: 13.9949934 percent

64 children: 9.9346754 percent

128 children: 7.0386092 percent

This is worrying. Does it mean that in a large population Jacks might drastically outnumber Jills?

No. As the population grows, the distribution assumes the shape of a normal bell curve concentrated near 1/2. So while the chance of *precise* parity drops, the chance that a large population will have approximately equal numbers of girls and boys actually increases, as we’d expect.

## “Constantly Mean”

The golden mean is quite absurd;

It’s not your ordinary surd.

If you invert it (this is fun!),

You’ll get itself, reduced by one;

But if increased by unity,

This yields its square, take it from me.

— Paul S. Bruckman, *The Fibonacci Quarterly*, 1977

## Both Sides Now

Bach’s “crab canon” rendered as a Möbius strip:

Bach and Handel were both blinded by the same oculist, John Taylor, “the poster child for 18th-century quackery,” according to University of Wisconsin ophthalmologist Daniel Albert. Bach probably died of a post-operative infection; Handel wrote the lyrics to *Samson* (“Total eclipse! No sun, no moon! / All dark amidst the blaze of noon!”) after Taylor’s botched cataract surgery.

Random Möbius anecdote: In 1957, B.F. Goodrich patented a half-twisted conveyor belt for carrying hot material such as cinders and foundry sand, “thereby permitting each face of the belt to cool during one half of the operating period.”

## Going Down

What would happen if you jumped into a tunnel that passed through the center of the earth? If you encountered no air resistance, dinosaurs, Mad Hatters, or Morlocks, you’d accelerate until you passed through the center at 18,000 mph, then slow as you ascended through the opposite hemisphere. At the far end you’d have just time to tip your hat to the surprised antipodeans before you fell home again, and you’d continue oscillating like this forever.

“If this shaft had its starting-point on one of the mountain plateaux of South America at an elevation of seven thousand feet,” wrote Camille Flammarion in 1909, “and if it issued at the sea-level at the other side, a man who had fallen into the shaft would arrive at the antipodes still travelling at such a speed that the spectators would see this strange projectile shot to a height of seven thousand feet into the air.”

On the other hand, if our straight tunnel connected two points that were not precise antipodes, then we could install a train powered by gravity — it would roll “downhill” on the first part of its journey, and momentum would carry it through the second (again neglecting air resistance and friction). Curiously, in all these cases the total trip would take the same length of time — about 42 minutes.

## Piece Work

Mathematician Henri Picciotto was visiting a local McDonald’s with his son when he noticed that Chicken McNuggets are served in boxes of 6, 9, and 20 nuggets. This means that you can easily buy, say, exactly 15 nuggets, but not 16. What’s the largest “non-McNugget” number?

He worked it out on a napkin: It’s 43. If you want any number of Chicken McNuggets larger than 43, you can get them by buying some combination of 6-, 9-, and 20-nugget boxes. But if you want exactly 43, you’re out of luck.

(That was in the 1980s. Today you can buy 4 McNuggets in a Happy Meal, which simplifies the problem. Perhaps McDonald’s was listening.)

## Misc

- Newton was born the year that Galileo died.
- Cole Porter’s summer home was called No Trespassing.
- 66339 = (6 × 6)
^{3}+ 3^{9} - Could you have had different parents?
- “A good conscience is a continual Christmas.” — Ben Franklin

UPDATE: The first item here is incorrect. The dates coincide only if one uses the Gregorian calendar to date Galileo’s death and the Julian to date Newton’s birth. The two events occurred 361 days apart, which puts them in separate years on both calendars. Apparently this is a very common error. (Thanks, Igor.)

## Snappy New Year

In *Insurmountable Simplicities* (2006), Roberto Casati points out that a traveler flying east may miss midnight — by entering a new time zone, he may jump from the 11:00 hour into the 12:00 hour without passing through midnight:

Mathematics and geography tell us that during the flight it will happen more than once that we reach the stroke of the hour without leaving the time zone we are in. If our flight lasts eight hours and the time difference between New York and Paris is six hours, this will happen at least twice, and at most eight times. But there is no guarantee that the stroke of midnight will be among these cases.

Thus, if it’s New Year’s Eve, an eastbound traveler may get no champagne.

## Frozen Fire

Lightning can fuse sand into curious rootlike tubes up to 5 meters long, called fulgurites. Because their shape records the path of the strike as it passes into the ground, they’re sometimes known as petrified lightning.

Lightning had a ruinous history before the introduction of Ben Franklin’s lightning rod. The campanile of St. Mark in Venice was destroyed three times over. In 1769, a bolt struck the tower of St. Nazaire in Brescia, whose magazine contained 100 tons of gunpowder. One-sixth of the town was destroyed, and 3,000 people died.

Compounding the harm was the disastrous belief that ringing bells during thunderstorms would allay lightning. In one 33-year period, lightning struck 386 church towers and killed 103 bell ringers.

Modern strikes are less dire. In 1919, Cleveland Indians pitcher Ray Caldwell was struck by lightning during a game against the Philadelphia Athletics. “It felt like a sandbag hit me,” he said. He refused to leave the game and pitched to Joe Dugan for the final out. The Indians won, 2-1.

## Bites and Pieces

Properly speaking, can the top half of an apple exist without the whole?

What is it the top half *of*?

## Crowded

Any set of 10 positive integers smaller than 100 will always contains two subsets with the same sum.

In any such group, the number of possible subsets (excluding the empty set) is 2^{10} – 1, or 1023. And the largest possible sum of any subset is 90 + 91 + … + 99 = 945. Hence, no matter which numbers are chosen, there will always be more subsets than possible sums, and some subsets (dozens, actually) must yield the same sum.

## Squeeze Play

A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?

## The Pizza Theorem

If you’re sharing a pizza with another person, there’s no need to cut it into precisely equal slices.

Make four cuts at equal angles through an arbitrary point and take alternate slices. You’ll both get the same amount of pizza.

Also: If a pizza has thickness *a* and radius *z*, then its volume is pi *z* *z* *a*.

## Misc

- Dorothy Parker named Alexander Woollcott’s apartment “Wit’s End.”
- Can you look at something and imagine it at the same time?
- 36850 = (3
^{6}+ 8) × 50 - AGNOSTIC is an anagram of COASTING.
- “The errors of a man are what make him really lovable.” — Goethe

## Skyline Trouble

In June 1978, a Princeton engineering student called structural engineer William LeMessurier with some worrying calculations. LeMessurier’s new Citicorp Tower, which had opened the previous year, was vulnerable to quartering winds — winds that blew from a 45-degree angle. On investigating, LeMessurier found also that the welded joints he had specified had been replaced with weaker bolted joints during construction. This meant that a strong wind could shear the bolts and topple a 59-story building into midtown Manhattan.

With hurricane season approaching, welders worked from 8 p.m. to 4 a.m. every night, reinforcing the building’s joints, and the Red Cross worked out an evacuation plan for the surrounding neighborhood. Because of a press strike at the time, many of these details came to light only 20 years later.

That year’s Hurricane Ella actually bore down on New York as the workers were finishing the job, but the storm veered out to sea before reaching the city. The welding was completed in October, and it’s now estimated that a storm strong enough to rock the tower will occur only once every 700 years.

## Tock

How fast does time pass? We have no way to measure this. We can reply, helplessly, that it passes at one second per second, but this is not a rate of change — 1 second divided by 1 second is 1. Not 1 *of* anything, just 1.

“‘One’ can be an answer, right or wrong, to the questions ‘How many children had Lady Macbeth?’, ‘How many Gods are there?’, and ‘How many minutes do sixty seconds make?’,” writes Notre Dame philosopher Peter van Inwagen. “‘One’ can never be an answer, not even a wrong one, to *any* other sort of question — including those questions that ask ‘how fast?’ or ‘at what rate?’ Therefore, if time is moving, it is not moving at any rate or speed.”

## Ceramic Geometry

This tiling pattern is sometimes referred to as Pythagorean because it can be construed to prove the Pythagorean theorem.

The red area is a right triangle. The square of its shorter side is equivalent to a green square, and the square of its longer side is equivalent to a yellow square.

One green and one yellow square can be cut up and reassembled to fit into one of the canted white squares, which is equivalent to the square of the red triangle’s hypotenuse. Hence a^{2} + b^{2} = c^{2}.

## Keeping Things Interesting

Each year, when the last flight of the summer field season departs the U.S. research station at the South Pole, the remaining staff gather to watch *The Thing*.

The next flight won’t arrive for eight months.

## Time and Again

Imagine that I first walk through Divinity Avenue, and then imagine that the powers governing the universe annihilate ten minutes of time with all that it contained, and set me back at the door of this hall just as I was before the choice was made. Imagine then that, everything else being the same, I now make a different choice and traverse Oxford Street. You, as passive spectators, look on and see the two alternative universes,–one of them with me walking through Divinity Avenue in it, the other with the same me walking through Oxford Street. Now, if you are determinists you believe one of these universes to have been from eternity impossible: you believe it to have been impossible because of the intrinsic irrationality or accidentality somewhere involved in it. But looking outwardly at these universes, can you say which is the impossible and accidental one, and which the rational and necessary one? I doubt if the most ironclad determinist among you could have the slightest glimmer of light on this point. In other words, either universe

after the factand once there would, to our means of observation and understanding, appear just as rational as the other.

— William James, “The Will to Believe,” 1896

## Healthcare Reform

The statement “You will recover from this illness” is either true or false. If it’s true, then it has been true for all eternity, and you’ll recover whether you summon a doctor or not.

If the statement is false, then it has always been false, and you will not recover even with a doctor’s aid.

So there is no point in calling a doctor.

(From Cicero’s *De Fato*.)

## A Good Man

Should other species be regarded as human? In 1779 Lord Monboddo proposed that orangutans should: They walk upright, use weapons, form societies, build shelters, and behave with “dignity and composure.” “If … such an Animal be not a Man, I should desire to know in what the essence of a Man consists, and what it is that distinguishes a Natural Man from the Man of Art?”

Thomas Love Peacock mocked this view in his 1817 novel *Melincourt*, in which a civilized orangutan (“Sir Oran Hout-ton”) is elected to Parliament. And an anonymous wag objected even to the satire:

The author of a novel lately written,

Entitled “Melincourt,”

(‘Tis very sweet and short),

Seems indeed by some wondrous madness bitten,

Thinking it good

To take his hero from the wood:

And though I own there’s nothing treasonable

In making ouran-outangs reasonable,

I really do not think he should

Go quite the length that he has done,

Whether for satire or for fun,

To make this creature an M.P.

As if mankind no wiser were than he.

However, those who’ve read it

Must give the author credit

For skill and ingenuity,

Although it have this monstrous incongruity.

But today Monboddo’s view is on the ascendancy. Harvard legal scholar Steven M. Wise argues that orangutans — as well as chimpanzees, bonobos, elephants, parrots, dolphins, and gorillas — deserve legal personhood. “Ancient philosophers claimed that all nonhuman animals had been designed and placed on this earth just for human beings,” he writes. “Ancient jurists declared that law had been created just for human beings. Although philosophy and science have long since recanted, the law has not.”