Quiz

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Gustave Flaubert posed this teasing problem to his sister Caroline in an 1841 letter:

Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?

He didn’t give an answer. Elsewhere he wrote, “To be stupid, selfish, and have good health are three requirements for happiness — though if stupidity is lacking, all is lost.”

Paint by Number

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When Mariner 4 flew past Mars in summer 1965, NASA scientists were eager to get their first close look at another planet. So rather than wait for their computers to render the probe’s data into a proper photograph, the employees in the agency’s telecommunications group mounted printed strips of data in a display panel and colored them by hand to create a rough visualization.

The hand-colored vista became the first image of Mars based on data collected by an interplanetary probe. They framed the finished image and presented it to agency director William H. Pickering.

https://photojournal.jpl.nasa.gov/catalog/PIA14033

Double Alphamagic Squares

In 1986 British electronics engineer Lee Sallows invented the alphamagic square:

alphamagic square 1

As in an ordinary magic square, each row, column, and long diagonal produces the same sum. But when the number in each cell is replaced by the length of its English name (25 -> TWENTY-FIVE -> 10), a second magic square is produced:

alphamagic square 2

Now British computer scientist Chris Patuzzo, who found the percentage-reckoned pangram that we covered here in November 2015, has created a double alphamagic square:

double alphamagic square 1

Each row, column, and long diagonal here totals 303370120164. If the number in each cell is replaced by the letter count of its English name (using “and” after “hundred,” e.g. ONE HUNDRED AND FORTY-EIGHT BILLION SEVEN HUNDRED AND TWENTY-EIGHT MILLION THREE HUNDRED AND SEVENTY-EIGHT THOUSAND THREE HUNDRED AND SEVENTY-EIGHT), then we get a new magic square, with a common sum of 345:

double alphamagic square 2

And this is itself an alphamagic square! Replace each number with the length of its name and you get a third magic square, this one with a sum of 60:

double alphamagic square 3

Chris has found 50 distinct doubly alphamagic squares, listed here. I suppose there must be some limit to this — is a triple alphamagic square even possible?

(Thanks, Chris and Lee.)

Nature Reading

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Image: Wikimedia Commons

In Germany, where modern forestry began, a curious new sort of literature arose in the 18th century:

Some enthusiast thought to go one better than the botanical volumes that merely illustrated the taxonomy of trees. Instead the books themselves were to be fabricated from their subject matter, so that the volume on Fagus, for example, the common European beech, would be bound in the bark of that tree. Its interior would contain samples of beech nuts and seeds; and its pages would literally be its leaves, the folios its feuilles.

That’s from Simon Schama’s Landscape and Memory, 1995. These xylotheques, or wood repositories, grew up throughout the developed world — the largest, now held by the U.S. Forest Service, houses 60,000 samples. “But the wooden books were not pure caprice, a nice pun on the meaning of cultivation,” Schama writes. “By paying homage to the vegetable matter from which it, and all literature, was constituted, the wooden library made a dazzling statement about the necessary union of culture and nature.”

Math Limericks

There was an old man who said, “Do
Tell me how I’m to add two and two!
I’m not very sure
That it does not make four,
But I fear that is almost too few.”

A mathematician confided
A Möbius strip is one-sided.
You’ll get quite a laugh
If you cut one in half,
For it stays in one piece when divided.

A mathematician named Ben
Could only count modulo ten.
He said, “When I go
Past my last little toe,
I have to start over again.”

By Harvey L. Carter:

‘Tis a favorite project of mine
A new value of π to assign.
I would fix it at 3,
For it’s simpler, you see,
Than 3.14159.

J.A. Lindon points out that 1264853971.2758463 is a limerick:

One thousand two hundred and sixty
four million eight hundred and fifty
three thousand nine hun-
dred and seventy one
point two seven five eight four six three.

From Dave Morice, in the November 2004 Word Ways:

A one and a one and a one
And a one and a one and a one
And a one and a one
And a one and a one
Equal ten. That’s how adding is done.

(From Through the Looking-Glass:)

‘And you do Addition?’ the White Queen asked. ‘What’s one and one and one and one and one and one and one and one and one and one?’

‘I don’t know,’ said Alice. ‘I lost count.’

‘She can’t do Addition,’ the Red Queen interrupted.

Two classics, one from Leigh Mercer:

\displaystyle \frac{12 + 144 + 20 + 3 \sqrt{4}}{7} + \left ( 5 \times 11 \right ) = 9^{2} + 0

A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more.

And another anonymous:

\displaystyle \int_{1}^{\sqrt[3]{3}}z^{2}dz \times \textup{cos} \frac{3\pi }{9} = \textup{ln} \sqrt[3]{e}

The integral z-squared dz
From one to the cube root of three
Times the cosine
Of three pi over nine
Equals log of the cube root of e.

UPDATE: Reader Jochen Voss found this on a blackboard at Warwick University:

If M’s a complete metric space
(and non-empty), it’s always the case:
If f’s a contraction
Then, under its action,
Exactly one point stays in place.

And Trevor Hawkes sent this:

A mathematician called Klein
Thought the Möbius strip was divine.
He said if you glue
The edges of two
You get a nice bottle like mine.

The Kate Bush Conjecture

Many thanks to reader Colin White for this:

In her 2005 song “π,” Kate Bush sings the number π to its 78th decimal place, then jumps abruptly to the 101st and finishes at the 137th.

The BBC’s More or Less advanced the “Kate Bush conjecture”: that the digits that Bush sings are contained somewhere in the decimal expansion of π — just not at the start.

The conjecture is true if π turns out to be a “normal” number, meaning essentially that all possible sequences of digits (of a given length) appear equally often in its expansion.

π hasn’t been proven to have this property, though it’s expected to be the case. So, for now, “The Kate Bush conjecture is plausible but unproven.”

Science Fiction

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For the writer of fantastic stories to help the reader to play the game properly, he must help him in every possible unobtrusive way to domesticate the impossible hypothesis. He must trick him into an unwary concession to some plausible assumption and get on with his story while the illusion holds. And that is where there was a certain slight novelty in my stories when first they appeared. Hitherto, except in exploration fantasies, the fantastic element was brought in by magic. Frankenstein even, used some jiggery-pokery magic to animate his artificial monster. There was trouble about the thing’s soul. But by the end of last century it had become difficult to squeeze even a momentary belief out of magic any longer. It occurred to me that instead of the usual interview with the devil or a magician, an ingenious use of scientific patter might with advantage be substituted. That was no great discovery. I simply brought the fetish stuff up to date, and made it as near actual theory as possible.

— H.G. Wells, June 1934 (from the H.G. Wells Scrapbook)

“Holes” and Factors

Here are the proper prime divisors of the first nine natural numbers (a proper prime divisor is a prime different from n that divides n evenly):

1: (none)
2: (none)
3: (none)
4: 2 × 2
5: (none)
6: 2 × 3
7: (none)
8: 2 × 2 × 2
9: 3 × 3

So, if we include repeated instances of a given factor:

  • 1, 2, 3, 5, and 7 have 0 proper prime divisors
  • 4, 6, and 9 have 2 proper prime divisors
  • 8 has 3 proper prime divisors

Mathematicians Ana Luzón and Manuel A. Morón of Universidad Politecnica de Madrid point out a coincidence: The numerals in each of these groups have the same basic shape — within each group it’s possible to transform one numeral into another by bending, shrinking, and expanding. So, for example, it’s possible to bend a numeral 1 made of clay into a 2 or a 7, but not into a 9 — we’re not allowed to poke a new hole in the clay or to affix one part of it to another.

Luzón and Morón write that if two of these nine numerals have the same number of proper prime divisors, then those two will “cut a sheet in the same number of pieces if you write them down with a scalpel.” And if the scalpel doesn’t cut the sheet into multiple pieces, then the number you’re writing is prime (except for 1).

Note: This works only if the numeral 4 is “closed” at the top, not open. So this post will make sense if you’re reading it on Futility Closet (which uses the “closed” font Georgia), but possibly not if you’re reading it in a different font elsewhere. Maybe this tells us how 4 “ought” to be written!

(Ana Luzón and Manuel A. Morón, “4 or 4? Mathematics or Accident?” Mathematics Magazine 75:4 [October 2002], 274.)

Math and Pancakes

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Image: Wikimedia Commons

If you apply one straight cut to a pancake, pretty clearly you’ll get 2 pieces. With two cuts, the most you can get is 4. What’s the greatest number you can produce with three cuts? If the cuts meet neatly in the center, you’ll get 6 pieces, but if you’re artfully sloppy you can make 7 (above). Charmingly, this leads us into the “lazy caterer’s sequence” — the maximum number of pieces you can produce with n straight cuts:

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, …

Generally it turns out that the maximum number for n cuts is given by the formula

\displaystyle p = \frac{n^{2} + n + 2}{2};

each number equals 1 plus a triangular number.

A related question is the pancake flipping problem. You’re presented with a spatula and an untidy stack of pancakes of varying sizes. You can insert the spatula at any point in the stack and flip all the pancakes above it. What’s the least number of flips required to sort the pancakes in order of size? Interestingly, no one has found a general answer. It’s possible to work out the solution for relatively small stacks (in which the number of pancakes is 1, 2, 3, …):

0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, …

But no one has found a formula that will tell how many flips will get the job done for a stack of any given size.

The problem has an interesting pedigree. Bill Gates worked on it at Harvard (PDF), and David X. Cohen, who went on to write for The Simpsons and Futurama, worked on a related problem at Berkeley in which the bottom of each pancake is burnt and the sort must be completed with the burnt sides facing down.

CCNY mathematician Jacob Goodman, who first hit on the pancake flipping problem while sorting folded towels for his wife, submitted it to the American Mathematical Monthly under the name Harry Dweighter (“harried waiter”). His household chores have produced at least one other publication: After some thoughtful work with a swivel-bladed vegetable peeler, he published “On the Largest Convex Polygon Contained in a Non-convex n-gon, Or How to Peel a Potato.”

(Thanks, Urzua.)