5 Math-Based Home Hacks That Will Make Your Life Easier

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iStock

Even those who like math may struggle to see how it applies to their everyday lives; even those who grant that mathematics underpins marvels from cybersecurity to moon landings may doubt the discipline’s relevance to matters mundane or domestic. Many problems routinely encountered around the house, however, do in fact benefit from mathematical methods and insights. Here’s a selection.

1. FOLDING A FITTED SHEET

To those who lack mathematically-inclined minds, mathematicians have out-of-this-world intelligence—and, with it, the ability to perform impossible feats. Folding a fitted sheet, for example.

“You should be able to figure out how to fold a fitted sheet,” an acquaintance once told Mathematical Association of America ambassador James Tanton. “It’s just topology, after all.” (Topology is the mathematical study of properties that are preserved under such deformations as stretching, crumpling, and bending, but with no tearing or gluing allowed.)

Thus goaded, Tanton brought his mathematical training to bear on the problem. Applying such tried-and-true strategies as working backwards and following your nose, he produced an instructional video (above) that will have you tidily storing elasticized bedclothes in no time. First, hold up the sheet so that the short sides are perpendicular to the floor, then stick your hands into the top two corners. Next, bring your hands together; hold the corners with one hand still inside the sheet and pull the outer corner over that hand. Lay the sheet on the table and attend to the messy side, tucking the inner corner inside the outer corner. Pick up the sheet by the corners and shake it, then lay it on the table again; once you've fixed any lingering messiness, the elastic should form an upside-down U-shape, and the sheet itself should be a rectangle. Looking at the upside-down U, fold the right side over to the left side, turn 90 degrees and fold in thirds. Finally, turn it 90 degrees and fold in thirds again, and voilà! A fitted sheet folded just as neatly as a flat sheet.

2. FLIPPING A MATTRESS

Unless it boasts a must-face-up pillow-top, a mattress can be placed on a bed frame four different ways. There are two possible sleep surfaces, each of which has two possible orientations (since one or the other short side must be at the head of the bed). For minimal wear, a mattress should spend equal time in each of the four configurations. But how is an absent-minded mattress owner to accomplish this? Is there a certain mattress maneuver that could be performed quarterly to cycle through the four arrangements?

Science writer Brian Hayes explored this question in his 2005 American Scientist article “Group Theory in the Bedroom.” Group theory is a branch of mathematics that’s handy for studying symmetry, and Hayes’s article offers an accessible introduction. Hayes ends up establishing, however, that there is no “golden rule of mattress flipping,” no maneuver one can mindlessly execute to hit each arrangement in turn.

But we're not doomed to a future of unevenly worn sleep surfaces. Hayes suggests that scrupulous sleepers do the following: Number the four mattress orientations 0, 1, 2, and 3, labeling each with a number in the corner closest to the righthand side of the head of the bed. Then, cycle through the orientations 0, 1, 2, 3, 0, 1, 2, 3, 0, etc., each quarter turning the mattress to position the next number in the upper right. Problem solved.

3. DIVIDING RENT

Suppose a handful of housemates must decide who will pay how much rent. They could just divide the total evenly, or perhaps base the division on the relative square footage of the various bedrooms. Experts in a field called "fair division," though, have a better way, one that can account for differing views on what’s valuable in a room—one roommate might crave natural light, while another would readily trade sunshine for a walk-in closet or a straight shot to the loo. The math-based method, which works thanks to a 1928 result called Sperner's Lemma, is also envy-free, meaning that no one will want to swap his room/rent payment pair for someone else's.

Mathematician Francis Su applied Sperner’s Lemma to rent partitioning in a 1999 paper [PDF]; The New York Times sketched the procedure in a 2014 article; and earlier this year “Mathologer” Burkard Polster explicated the Times piece in a 15-minute video. Online tools such as this one, however, allow would-be housemates to generate everybody’s-happy rent divisions just by entering number of housemates and total rent and then each answering a series of questions of the form “If the rooms have the following prices, which room would you choose?” As you go through the calculator, it narrows down the price range each roommate finds acceptable for each room and then finds a region where all the roommates have a room at a price they consider fair.

Users must, of course, keep their expectations realistic. If two people want the same room and are willing to pay anything for it—even if that means the other rooms are free—then the calculator won’t work. But there are also sociological concerns. “It is unfortunately beyond the scope of any algorithm,” cautions the rent calculator’s disclaimer, “to keep you from envying your roommate’s job, sex life or wardrobe—or save you from buyer’s remorse.”

4. CUTTING A CAKE

Portion envy can poison a party. So a host doling out any continuous foodstuff—cake, pizza, a 6-foot submarine sandwich—would do well to heed insights gleaned from the study of fair division.

If two people are sharing a dessert or an entree, of course, the problem is simple enough: Person A divides the dish into two portions she deems equal—maybe the piece of cake with the buttercream rose is smaller than the one without, to account for A’s taste for that decoration—and then Person B claims the portion she prefers. This division, like the rent partitioning discussed above, is envy-free: Neither person would rather have the other’s share.

Two-party division has been understood since biblical times, and a method of producing an envy-free division among three parties has been known for more than 50 years (see this article for an illustrated explanation of the cutting and trimming involved). A comparable procedure for more than three parties proved elusive until 2016, however, when computer scientists Simon Mackenzie and Haris Aziz outlined “a discrete and bounded envy-free cake cutting protocol for four agents” [PDF]. The pair subsequently adapted their protocol to cover any number of agents [PDF], but there’s a catch: Dividing a cake among even a handful of would-be eaters can require more steps than there are atoms in the universe. So hosts who want to serve their guests before staleness sets in may need to risk a little envy.

5. MOVING A SOFA

Anyone with 1) an L-shaped hallway leading from door to living room and 2) a fondness for multi-person upholstered seating may face the so-called “moving sofa problem.” Posed (more abstractly) in 1966 by mathematician Leo Moser, the problem asks for the largest sofa (in terms of seating area) that can be maneuvered around a right-angled corner without lifting, squishing, or tilting.

A square sofa with the same width—1, say—as the hallway could fit by scooting into the corner and then changing direction, but would have an area of only 1. A semicircular sofa with radius 1 would arc around nicely by using the curve to swing around the inside corner and increase the area to about 1.57. Mathematicians John Hammersley and Joseph Gerver devised corner-clearing sofa shapes, both reminiscent of old telephone handsets, with areas approximately 2.2074 and 2.2195, respectively. No one is sure that a couch made to Gerver’s specifications—the outline of the seating area comprises no fewer than 18 pieces—would be the largest one capable of rounding the corner, but it’s the best bet to date.

But what if a sofa must turn twice, once to the right and once to the left, to reach its final resting place? Mathematician Dan Romik puzzled over this variation on the moving sofa problem in recent years, and discovered a two-lobed “ambidextrous sofa” shape with area about 1.64495. The Romik Ambiturner may be the largest possible, but nothing has been proven yet. Interested readers can browse (animated!) sofa shapes on Romik's website.

11 Dothraki Words and Phrases Every Game of Thrones Fan Should Know

Helen Sloan, HBO
Helen Sloan, HBO

You know the words khal and khaleesi, but consider working these other words and phrases from the Dothraki language—which was created by linguist David J. Peterson, and featured in Living Language Dothraki—into your vocabulary before the final season of Game of Thrones premieres on April 14, 2019.

1. M’athchomaroon!

The Dothraki way of saying hi, this word—which can also be shortened to M’ach! or M’ath!—translates to “With respect.” To say hello to a group of non-Dothrakis, you would use the phrase Athchomar chomakea, which literally translates to “Respect to those that are respectful.” Fonas chek, which translates to "hunt well," is one way to say goodbye.

2. San athchomari yeraan!

Peterson writes that the Dothraki have no word for “thank you.” Instead, use this phrase, which literally translates to “a lot of honor to you!” but basically means “much respect!”

3. Fichas jahakes moon!

These are Dothraki fighting words, meant to encourage the warriors in their khalasar (or Dothraki group). This phrase means “get him!” but literally translates to “Take his braid”—which makes sense, since Dothrakis cut off their braids after a defeat. A Dothraki who wins a lot of battles is a lajak haj, or “strong warrior.”

4. And 5. Yer shekh ma shieraki anni and Yer jalan atthirari anni

Jason Momoa and Emilia Clarke in Game of Thrones
Helen Sloan, HBO

Both of these phrases—the first said by a male, the second by a female—mean “you are my loved one,” but they literally translate to phrases well-known to Game of Thrones fans: “You are my sun and stars” and “You are the moon of my life.” As Peterson notes, “these expressions come from Dothraki mythology, in which the sun is the husband of the moon.”

6. Anha dothrak adakhataan

Peterson writes that “as a result of the importance of horses to Dothraki culture, there are many idiomatic expressions related to horses and riding.” This phrase is best used before a meal: It means “I’m about to eat,” and literally translates to “I ride to eating.” If you were Dothraki, you’d likely be eating fresh horsemeat (gavat) and drinking mare’s milk (lamekh ohazho, which is often just shortened to lamekh).

7. Hrazef

This is Dothraki for horse, and there are many other words relating to horses in the language. A good one to know is the word for the great stallion, a.k.a., “the deity worshipped by the Dothraki”: vezhof.

8. Addrivat

Joseph Naufahu, Tamer Hassan, Emilia Clarke, Elie Haddad, Darius Dar Khan, and Diogo Sales in Game of Thrones
HBO

If there’s one thing the Dothraki are very good at, it’s killing, and they have multiple words for the deed. This is a verb meaning “to kill,” and literally translates to “to make something dead.” Both Ds are pronounced. It’s used, according to Peterson, “when the killer is a sentient being.” (Drozhat is used when a person is killed by an animal or an inanimate object, "like a fallen rock," Peterson writes.)

9. Asshekhqoyi vezhvena!

The next time your friend or loved one is celebrating another year around the sun, use this Dothraki phrase, which means “happy birthday” but literally translates to “[Have] a great blood-day!”

10. Zhavorsa or Zhavvorsa

Dothraki for dragon. Finne zhavvorsa anni? means “Where are my dragons?” This word might not be super applicable in everyday life, so jano—the Dothraki word for dog or dogs—is probably more appropriate.

11. Vorsa

Dracarys—a.k.a., what Dany says to Drogon to get him to let loose—is the High Valyrian word for dragonfire. It's unclear if the Dothraki have a word for dragonfire, but the word for fire is vorsa. Sondra, meanwhile, is their word for obsedian—or, as it's called on Game of Thrones, dragonglass.

For more information on the Dothraki language and culture, pick up Living Language Dothraki: A Conversational Language Course Created by David J. Peterson Based on the Hit Original HBO Series Game of Thrones at Amazon.

13 Facts About the Oxford English Dictionary

iStock.com/GCShutter
iStock.com/GCShutter

This year marks the 135th birthday of the Oxford English Dictionary (though the eminent reference book is hardly looking its age). As the English language continues to evolve, the dictionary has flourished and regularly added new words such as nothingburger, prepper, idiocracy, and fam. Get to know it better.

1. The Oxford English Dictionary (OED) was built on volunteer labor.

When the London Philological Society came up with the idea for a new dictionary of the English language in 1857, the editors decided it was necessary to enlist the help of the public and asked avid readers to send examples of sentences that could illuminate the meanings of different words. Every day, volunteers mailed thousands of “quotation slips” from books, newspapers, and magazines. By the time the first edition was published, more than 2000 volunteers had assisted the editors in its completion.

2. It took more than 70 years to complete the first edition of the OED.

Originally, the Philological Society predicted that the dictionary would take about 10 years to complete. Twenty-seven years later, the editors had successfully reached the word ant. Knowing it would be a while until a completed book was ready, they began publishing unbound editions of the work-in-progress in 1884. The first full volume was eventually published in 1928, more than 70 years after the society first came up with the idea.

3. The OED started out messy. Very messy.

Frederick Furnivall, one of the dictionary’s founders, was a visionary—but that vision did not extend to his organizational skills. Under his stewardship as editor, the dictionary was a mess. Quotation slips were stuffed haphazardly into bags and went missing. All of the words starting with “Pa” went AWOL for 12 years and were eventually discovered in Ireland. Slips for the letter “G” were nearly burned with somebody’s trash. All of the entries for the letter “H” somehow turned up in Italy.

4. OED co-founder Frederick Furnivall was a controversial figure.

After founding a controversy-riddled Shakespeare Society, Furnivall fell into a six-year feud with the poet Algernon Charles Swinburne. Swinburne (whose mastery of the English language earned him six nominations for the Nobel Prize in Literature) mocked Furnivall’s club by calling it “Fartiwell and Co.” and “The Sh*tspeare Society.” Furnivall reached into his bag o' insults and said that Swinburne had, “the ear of a poetaster, hairy, thick and dull.”

5. Dr. James Murray helped the OED clean up its act.

Sir James Murray in his Scriptorium
Sir James Murray in his Scriptorium
Oxford English Dictionary // Public Domain

Dr. James Murray, a philologist, took the helm as the dictionary’s principal editor in 1879 and remained in that position for the rest of his life (he died in 1915). Murray was a linguistic superstar; he was proficient in Italian, French, Catalan, Spanish, Latin, Dutch, German, Flemish, and Danish and also had a solid grasp of Portuguese, Vaudois, Provençal, Celtic, Slavonic, Russian, Persian, Achaemenid Cuneiform, Sanskrit, Hebrew, Syriac, Aramaic Arabic, Coptic, and Phoenician.

6. Murray built a shed for the OED's editors to work in.

In 1885, to better organize the dictionary, Murray constructed a sunken shed made of corrugated iron to house the editors and their precious quotation slips. Called the “Scriptorium,” this linguistic workshop contained 1029 pigeonholes that allowed Murray and his subeditors to arrange, sort, and file more than 1000 quotation slips each day. 

6. Only one word is known to have gone missing.

Only one quotation slip—containing the word bondmaid—is known to have been lost. (It fell down behind some books and the editors never noticed.) Murray was deeply embarrassed by his failure to include the word in the dictionary. “[N]ot one of the 30 people (at least) who saw the work at various stages between MS. and electrotyped pages noticed the omission,” he said. “The phenomenon is absolutely inexplicable.” The word was officially introduced in a 1933 supplement.

7. One of the OED’s most prolific contributors was a murderer confined to an insane asylum.

One volunteer who provided the OED with countless quotation slips was William C. Minor, a schizophrenic who was incarcerated at the Broadmoor Insane Asylum in Berkshire, England, after he fatally shot a man he (erroneously) believed had broken into his room. According to Murray, Minor was the dictionary’s second most prolific contributor, even outdoing members of the full-time staff.

8. J.R.R. Tolkien contributed to the OED, too.

In 1919 and 1920, J.R.R. Tolkien worked for the dictionary, where he studied the etymology of Germanic words beginning with the letter W, composing drafts for words like waggle and wampum. "I learned more in those two years than in any other equal period of my life,” Tolkien later said. (Years later, Tolkien spoofed his editors in a comic fable called Farmer Giles of Ham.)

9. The longest entry in the OED is for a three-letter word.

The most complicated word in the Oxford English Dictionary? Set. In the dictionary’s 1989 edition, the three-letter word contains 430 senses (that is, shades of meaning) and requires a 60,000-word definition. Other short words with endless definitions? Run (396 senses), go (368 senses), and take (343 senses).

10. The most popular edition of the OED was impossible to read with the naked eye.

Originally, the OED had a limited audience. Not only was a set of books expensive, it was also bulky and took up an entire bookshelf. In 1971, the Oxford University Press decided to publish a smaller, complete version that compressed nine pages into one. The text was so tiny that the two-volume book came with a magnifying glass. It quickly became one of the bestselling dictionaries on the market.

11. Digitizing the OED took a lot of work.

In the late 1980s, it took more than 120 typists, 55 proofreaders, and a total of 67 million keystrokes to digitize the entire contents of the Oxford English Dictionary. The process took 18 months.

12. Shakespeare isn’t the OED's most quoted source.

The OED's most quoted source is, in fact, the British daily newspaper The Times, which has 42,840 quotations (nearly 10,000 more than William Shakespeare). Coming in third and fourth are the Scottish novelist Walter Scott and the Philosophical Transactions of the Royal Society of London, respectively. When it comes to coining and introducing new words, Shakespeare isn’t first in that arena either; that honor belongs to Geoffrey Chaucer.

13. The last word in the OED is totally buggy.

Each year, about 2000 to 5000 new words, senses, and subentries are added to the Oxford English Dictionary. For years, the last word in the book was zynthum, a type of malty beer made in ancient Egypt. But in 2017, zynthum was usurped by zyzzyva, a type of South African weevil.

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