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

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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.

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10 Biting Facts About Snapping Turtles
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Here in the Americas, lake monster legends are a dime a dozen. More than a few of them were probably inspired by these ancient-looking creatures. In honor of World Turtle Day, here are 10 things you might not have known about snapping turtles.

1. THE COMMON SNAPPING TURTLE IS NEW YORK'S OFFICIAL STATE REPTILE.

Elementary school students voted to appoint Chelydra serpentina in a 2006 statewide election. Weighing as much as 75 pounds in the wild (and 86 in captivity), this hefty omnivore’s natural range stretches from Saskatchewan to Florida.

2. ALLIGATOR SNAPPING TURTLES CAN BE LARGE. (VERY LARGE.)

An alligator snapping turtle
NorbertNagel, Wikimedia Commons // CC BY-SA 3.0

Utterly dwarfing their more abundant cousin, alligator snappers (genus: Macrochelys) are the western hemisphere’s biggest freshwater turtles. The largest one on record, a longtime occupant of Chicago’s Shedd Aquarium, weighed 249 pounds.  

A monstrous 403-pounder was reported in Kansas during the Great Depression, though this claim was never confirmed.  

3. COMMON SNAPPERS HAVE LONGER NECKS AND SPIKIER TAILS.

Alligator snappers also display proportionately bigger heads and noses plus a trio of tall ridges atop their shells. Geographically, alligator snapping turtles are somewhat restricted compared to their common relatives, and are limited mainly to the southeast and Great Plains.

4. BOTH VARIETIES AVOID CONTACT WITH PEOPLE.

If given the choice between fight and flight, snapping turtles almost always distance themselves from humans. The animals spend the bulk of their lives underwater, steering clear of nearby Homo sapiens. However, problems can arise on dry land, where the reptiles are especially vulnerable. Females haul themselves ashore during nesting season (late spring to early summer). In these delicate months, people tend to prod and handle them, making bites inevitable.

5. YOU REALLY DON'T WANT TO GET BITTEN BY ONE. 

Snapping turtle jaw strength—while nothing to sneeze at—is somewhat overrated. Common snapping turtles can clamp down with up to 656.81 newtons (N) of force, though typical bites register an average of 209 N. Their alligator-like cousins usually exert 158 N. You, on the other hand, can apply 1300 N between your second molars.

Still, power isn’t everything, and neither type of snapper could latch onto something with the crushing force of a crocodile’s mighty jaws. Yet their sharp beaks are well-designed for major-league shearing. An alligator snapping turtle’s beak is capable of slicing fingers clean off and (as the above video proves) obliterating pineapples.

Not impressed yet? Consider the following. It’s often said that an adult Macrochelys can bite a wooden broom handle in half. Intrigued by this claim, biologist Peter Pritchard decided to play MythBuster. In 1989, he prodded a 165-pound individual with a brand new broomstick. Chomp number one went deep, but didn’t quite break through the wood. The second bite, though, finished the job.

6. SCIENTISTS RECENTLY DISCOVERED THAT THERE ARE THREE SPECIES OF ALLIGATOR SNAPPING TURTLES.

A 2014 study trisected the Macrochelys genus. For over a century, naturalists thought that there was just a single species, Macrochelys temminckii. Closer analysis proved otherwise, as strong physical and genetic differences exist between various populations. The newly-christened M. suwanniensis and M. apalachicolae are named after their respective homes—namely, the Suwannee and Apalachicola rivers. Further west, good old M. temminckii swims through the Mobile and the Mississippi.

7. THANKS TO A 19TH CENTURY POLITICAL CARTOON, COMMON SNAPPING TURTLES ARE ALSO KNOWN AS "OGRABMES." 

Snapping turtle cartoon
Urban~commonswiki via Wiki Commons // CC BY PD-US

Drawn by Alexander Anderson, this piece skewers Thomas Jefferson’s signing of the unpopular Embargo Act. At the president’s command, we see a snapping turtle bite some poor merchant’s hind end. Agitated, the victim calls his attacker “ograbme”—“embargo” spelled backwards.

8. ALLIGATOR SNAPPERS ATTRACT FISH WITH AN ORAL LURE …

You can’t beat live bait. Anchored to the Macrochelys tongue is a pinkish, worm-like appendage that fish find irresistible. Preferring to let food come to them, alligator snappers open their mouths and lie in wait at the bottoms of rivers and lakes. Cue the lure. When this protrusion wriggles, hungry fish swim right into the gaping maw and themselves become meals.

9.  … AND THEY FREQUENTLY EAT OTHER TURTLES. 


Complex01, WikimediaCommons

Alligator snappers are anything but picky. Between fishy meals, aquatic plants also factor into their diet, as do frogs, snakes, snails, crayfish, and even relatively large mammals like raccoons and armadillos. Other shelled reptiles are fair game, too: In one Louisiana study, 79.82% of surveyed alligator snappers had turtle remains in their stomachs.

10. YOU SHOULD NEVER PICK A SNAPPER UP BY THE TAIL.

Ideally, you should leave the handling of these guys to trained professionals. But what if you see a big one crossing a busy road and feel like helping it out? Before doing anything else, take a few moments to identify the turtle. If it’s an alligator snapper, you’ll want to grasp the lip of the upper shell (or “carapace”) in two places: right behind the head and right above the tail.

Common snappers demand a bit more finesse (we wouldn’t want one to reach back and nip you with that long, serpentine neck). Slide both hands under the hind end of the shell, letting your turtle’s tail dangle between them. Afterwards, clamp down on the carapace with both thumbs.

Please note that lifting any turtle by the tail can permanently dislocate its vertebrae. Additionally, remember to move the reptile in the same direction that it’s already facing. Otherwise, your rescue will probably turn right back around and try to cross the road again later. 

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10 Things You Might Not Know About Tina Fey
Jenny Anderson, Getty Images for Tony Awards Productions
Jenny Anderson, Getty Images for Tony Awards Productions

Tina Fey has transformed modern comedy more than just about anyone else. From the main stage of Second City to the writer’s room of SNL to extremely fetch comedy blockbusters, Elizabeth Stamatina Fey has built a national stage with a dry, eye-popping sarcasm and political satire where no one is safe. She has a slew of Emmys, Golden Globes, SAG, PGA, and WGA awards to prove it—plus a recent Tony nomination (her first). But, more importantly, she’s the closest thing we have to a national comic laureate.

Here are 10 facts about a fantastically blorft American icon.

1. SHE DID A BOOK REPORT ON COMEDY WHEN SHE WAS 11.

Fey got a very early start in comedy, watching a lot of The Mary Tyler Moore Show, Bob Newhart, and Norman Lear shows as a kid. Her father and mother sneaked her in to see Young Frankenstein and would let her stay up late to watch The Honeymooners. So it’s no surprise that she chose comedy as the subject of a middle school project. The only book she could get her hands on was Joe Franklin’s Encyclopedia of Comedians, but at least she made a friend. "I remember me and one other girl in my 8th grade class got to do an independent study because we finished the regular material early, and she chose to do hers on communism, and I chose to do mine on comedy," Fey told The A.V. Club. "We kept bumping into each other at the card catalog."

2. THE SCAR ON HER FACE CAME FROM A BIZARRE ATTACK THAT OCCURRED WHEN SHE WAS A CHILD.

Fey’s facial scar had been recognizable but unexplained for years until a profile in Vanity Fair revealed that the mark on her left cheek came from being slashed by a strange man when she was five years old. “She just thought somebody marked her with a pen,” her husband Jeff Richmond said. Fey wrote in Bossypants that it happened in an alleyway behind her Upper Darby, Pennsylvania, home when she was in kindergarten.

3. HER FIRST TV APPEARANCE WAS IN A BANK COMMERCIAL.

Saturday Night Live hired Fey as a writer in 1997. In 1995 she had the slightly more glamorous job of pitching Mutual Savings Bank with a radical floral applique vest and a handful of puns on the word “Hi.” In a bit of life imitating art, just as Liz Lemon’s 1-900-OKFACE commercial was unearthed and mocked on 30 Rock, the internet discovered Fey’s stint awkwardly cheering on high interest rates a few years ago and had a lot to say about her '90s hair.

4. SHE WAS THE FIRST WOMAN TO BE NAMED HEAD WRITER OF SNL.

Four years after that commercial and two after she joined Saturday Night Live’s writing staff, Fey earned a promotion to head writer. Up until that point, the head writers were named Michael, Herb, Bob, Jim, Steve. You get the picture. She acted as head writer for six seasons until moving on to write and executive produce 30 Rock. Since her departure, two more women (Paula Pell and Sara Schneider) have been head writers for the iconic show.

5. SHE’S THE YOUNGEST MARK TWAIN PRIZE WINNER.

Established in 1998, the Kennedy Center’s hilarious honor has mostly been awarded to funny people in the twilight of their careers. Richard Pryor was the first recipient, and comedians who made their marks decades prior like Lily Tomlin, Whoopi Goldberg, and George Carlin followed. Fey earned the award in 2010 when she was 40 years old, and the age of her successors (Carol Burnett, Bill Murray, Eddie Murphy, David Letterman ...) signals that she may hold the title of youngest recipient for some time.

6. SHE WROTE SATIRE FOR HER HIGH SCHOOL NEWSPAPER.

Fey was an outstanding student who was involved in choir, drama, and tennis, and co-edited the school’s newspaper, The Acorn. She also wrote a satirical column addressing “school policy and teachers” under the pun-tastic pseudonym “The Colonel.” Fey also recalled getting in trouble because she tried to make a pun on the phrase “annals of history.” Cheeky.

7. SHE MADE HER RAP DEBUT WITH CHILDISH GAMBINO ON "REAL ESTATE."

Donald Glover (a.k.a. Childish Gambino) first gained notice as a member of Derrick Comedy in college, and Fey hired him at the age of 23 to write for 30 Rock. Before jumping from that show to Community, Glover put out his first mixtape under his stage name. After releasing his debut album, Camp, in 2011, Gambino dropped a sixth mixtape called Royalty that featured Fey rapping on a song called “Real Estate.” “My president is black, and my Prius is blue!"

8. SHE VOICED PRINCESSES IN A BELOVED PINBALL GAME.

Between the bank commercial and Saturday Night Live, Fey has an intriguing credit on her resume: the arcade pinball machine “Medieval Madness.” Most of the game’s Arthurian dialogue was written by Second City members Scott Adsit (Pete Hornberger on 30 Rock) and Kevin Dorff, who pulled in fellow Second City castmate Fey to voice for an “Opera Singer” princess, Cockney-speaking princesses, and a character with a southern drawl. (You can hear some of the outtakes here.)

9. SHE USED MEAN GIRLS TO PUSH BACK AGAINST STEREOTYPES OF WOMEN IN MATH.

Tina Fey and Lindsay Lohan in 'Mean Girls' (2004)
Paramount Home Entertainment

There’s a ton of interesting trivia about Mean Girls, Fey’s first foray into feature film screenwriting. She bid on the rights to Rosalind Wiseman’s book that inspired the movie without realizing it didn’t have a plot. She initially wrote a large part for herself but kept whittling it down to focus on the teenagers, and her first draft was “for sure R-rated.” Fey also chose to play a math teacher to fight prejudice. “It was an attempt on my part to counteract the stereotype that girls can’t do math. Even though I did not understand a word I was saying.” Fey used a friend’s calculus teacher boyfriend’s lesson plans in the script.

10. SHE SET UP A SCHOLARSHIP IN HER FATHER’S NAME TO HELP VETERANS.

Fey’s father Donald was a Korean War veteran who also studied journalism at Temple University. When he died in 2015, Fey and her brother Peter founded a memorial scholarship in his name that seeks to aid veterans who want to study journalism at Temple.

"He was really inspiring," Fey said. "A lot of kids grow up with dreams of doing those things and their parents are fearful and want them to get a law degree and have things to fall back on, but he and our mom always encouraged us to pursue whatever truly interested us." Fey also supports Autism Speaks, Mercy Corps, Love Our Children USA, and other charities.

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