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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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Echo Bridge Home Entertainment
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15 Must-See Holiday Horror Movies
Echo Bridge Home Entertainment
Echo Bridge Home Entertainment

Families often use the holidays as an excuse to indulge in repeat viewings of Planes, Trains and Automobiles and Elf. But for a certain section of the population, the yuletide is all about horror. Although it didn’t truly emerge until the mid-1970s, “holiday horror” is a thriving subgenre that often combines comedy to tell stories of demented Saint Nicks and lethal gingerbread men. If you’ve never seen Santa slash someone, here are 15 movies to get you started.

1. THANKSKILLING (2009)

Most holiday horror movies concern Christmas, so ThanksKilling is a bit of an anomaly. Another reason it’s an anomaly? It opens in 1621, with an axe-wielding turkey murdering a topless pilgrim woman. The movie continues on to the present-day, where a group of college friends are terrorized by that same demon bird during Thanksgiving break. It’s pretty schlocky, but if Turkey Day-themed terror is your bag, make sure to check out the sequel: ThanksKilling 3. (No one really knows what happened to ThanksKilling 2.)

2. BLACK CHRISTMAS (1974)

Fittingly, the same man who brought us A Christmas Story also brought us its twisted cousin. Before Bob Clark co-wrote and directed the 1983 saga of Ralphie Parker, he helmed Black Christmas. It concerns a group of sorority sisters who are systematically picked off by a man who keeps making threatening phone calls to their house. Oh, and it all happens during the holidays. Black Christmas is often considered the godfather of holiday horror, but it was also pretty early on the slasher scene, too. It opened the same year as The Texas Chainsaw Massacre and beat Halloween by a full four years.

3. SILENT NIGHT, DEADLY NIGHT (1984)

This movie isn’t about Santa Claus himself going berserk and slaughtering a bunch of people. But it is about a troubled teen who does just that in a Santa suit. Billy Chapman starts Silent Night, Deadly Night as a happy little kid, only to witness a man dressed as St. Nick murder his parents in cold blood. Years later, after he has grown up and gotten a job at a toy store, he conducts a killing spree in his own red-and-white suit. The PTA and plenty of critics condemned the film for demonizing a kiddie icon, but it turned into a bona fide franchise with four sequels and a 2012 remake.

4. RARE EXPORTS: A CHRISTMAS TALE (2010)

This Finnish flick dismantles Santa lore in truly bizarre fashion, and it’s not easy to explain in a quick plot summary. But Rare Exports involves a small community living at the base of Korvatunturi mountain, a major excavation project, a bunch of dead reindeer, and a creepy old naked dude who may or may not be Santa Claus. Thanks to its snowy backdrop, the movie scored some comparisons to The Thing, but the hero here isn’t some Kurt Russell clone with equally feathered hair. It’s a bunch of earnest kids and their skeptical dads, who all want to survive the holidays in one piece.

5. TO ALL A GOODNIGHT (1980)

To All a Goodnight follows a by-now familiar recipe: Add a bunch of young women to one psycho dressed as Santa Claus and you get a healthy dose of murder and this 1980 slasher flick. Only this one takes place at a finishing school. So it’s fancier.

6. KRAMPUS (2015)

Although many Americans are blissfully unaware of him, Krampus has terrorized German-speaking kids for centuries. According to folklore, he’s a yuletide demon who punishes naughty children. (He’s also part-goat.) That’s some solid horror movie material, so naturally Krampus earned his own feature film. In the movie, he’s summoned because a large suburban family loses its Christmas cheer. That family has an Austrian grandma who had encounters with Krampus as a kid, so he returns to punish her descendants. He also animates one truly awful Jack-in-the-Box.

7. THE GINGERDEAD MAN (2005)

“Eat me, you punk b*tch!” That’s one of the many corny catchphrases spouted by the Gingerdead Man, an evil cookie possessed by the spirit of a convicted killer (played by Gary Busey). The lesson here, obviously, is to never bake.

8. JACK FROST (1997)

No, this isn’t the Michael Keaton snowman movie. It’s actually a holiday horror movie that beat that family film by a year. In this version, Jack Frost is a serial killer on death row who escapes prison and then, through a freak accident, becomes a snowman. He embarks on a murder spree that’s often played for laughs—for instance, the cops threaten him with hairdryers. But the comedy is pretty questionable in the infamous, and quite controversial, Shannon Elizabeth shower scene.

9. ELVES (1989)

Based on the tagline—“They’re not working for Santa anymore”—you’d assume this is your standard evil elves movie. But Elves weaves Nazis, bathtub electrocutions, and a solitary, super grotesque elf into its utterly absurd plot. Watch at your own risk.

10. SINT (2010)

The Dutch have their own take on Santa, and his name is Sinterklaas. Sinterklaas travels to the Netherlands via steamship each year with his racist sidekick Zwarte Piet. But otherwise, he’s pretty similar to Santa. And if Santa can be evil, so can Sinterklaas. According to the backstory in Sint (or Saint), the townspeople burned their malevolent bishop alive on December 5, 1492. But Sinterklaas returns from the grave on that date whenever there’s a full moon to continue dropping bodies. In keeping with his olden origins, he rides around on a white horse wielding a golden staff … that he can use to murder you.

11. SANTA’S SLAY (2005)

Ever wonder where Santa came from? This horror-comedy claims he comes from the worst possible person: Satan. The devil’s kid lost a bet many years ago and had to pretend to be a jolly gift-giver. But now the terms of the bet are up and he’s out to act like a true demon. That includes killing Fran Drescher and James Caan, obviously.

12. ALL THROUGH THE HOUSE (2015)

Another Santa slasher is on the loose in All Through the House, but the big mystery here is who it is. This villain dons a mask during his/her streak through suburbia—and, as the genre dictates, offs a bunch of promiscuous young couples along the way. The riddle is all tied up in the disappearance of a little girl, who vanished several years earlier.

13. CHRISTMAS EVIL (1980)

Several years before Silent Night, Deadly Night garnered protests for its anti-Kringle stance, Christmas Evil put a radicalized Santa at the center of its story. The movie’s protagonist, Harry Stadling, first starts to get weird thoughts in his head as a kid when he sees “Santa” (really his dad in the costume) groping his mom. Then, he becomes unhealthily obsessed with the holiday season, deludes himself into thinking he’s Santa, and goes on a rampage. The movie is mostly notable for its superfan John Waters, who lent commentary to the DVD and gave Christmas Evil some serious cult cred.

14. SANTA CLAWS (1996)

If you thought this was the holiday version of Pet Sematary, guess again. The culprit here isn’t a demon cat in a Santa hat, but a creepy next-door neighbor. Santa Claws stars B-movie icon Debbie Rochon as Raven Quinn, an actress going through a divorce right in the middle of the holidays. She needs some help caring for her two girls, so she seeks out Wayne, her neighbor who has an obsessive crush on her. He eventually snaps and dresses up as Santa Claus in a ski mask. Mayhem ensues.

15. NEW YEAR’S EVIL (1980)

Because the holidays aren’t over until everyone’s sung “Auld Lang Syne,” we can’t count out New Year’s Eve horror. In New Year’s Evil, lady rocker Blaze is hosting a live NYE show. Everything is going well, until a man calls in promising to kill at midnight. The cops write it off as a prank call, but soon, Blaze’s friends start dropping like flies. Just to tie it all together, the mysterious murderer refers to himself as … “EVIL.”

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The American Museum of Natural History
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10 Surprising Ways Senses Shape Perception
The American Museum of Natural History
The American Museum of Natural History

Every bit of information we know about the world we gathered with one of our five senses. But even with perfect pitch or 20/20 vision, our perceptions don’t always reflect an accurate picture of our surroundings. Our brain is constantly filling in gaps and taking shortcuts, which can result in some pretty wild illusions.

That’s the subject of “Our Senses: An Immersive Experience,” a new exhibition at the American Museum of Natural History in New York City. Mental Floss recently took a tour of the sensory funhouse to learn more about how the brain and the senses interact.

1. LIGHTING REVEALS HIDDEN IMAGES.

Woman and child looking at pictures on a wall

Under normal lighting, the walls of the first room of “Our Senses” look like abstract art. But when the lights change color, hidden illustrations are revealed. The three lights—blue, red, and green—used in the room activate the three cone cells in our eyes, and each color highlights a different set of animal illustrations, giving the viewers the impression of switching between three separate rooms while standing still.

2. CERTAIN SOUNDS TAKE PRIORITY ...

We can “hear” many different sounds at once, but we can only listen to a couple at a time. The AMNH exhibit demonstrates this with an audio collage of competing recordings. Our ears automatically pick out noises we’re conditioned to react to, like an ambulance siren or a baby’s cry. Other sounds, like individual voices and musical instruments, require more effort to detect.

3. ... AS DO CERTAIN IMAGES.

When looking at a painting, most people’s eyes are drawn to the same spots. The first things we look for in an image are human faces. So after staring at an artwork for five seconds, you may be able to say how many people are in it and what they look like, but would likely come up short when asked to list the inanimate object in the scene.

4. PAST IMAGES AFFECT PRESENT PERCEPTION.

Our senses often are more suggestible than we would like. Check out the video above. After seeing the first sequence of animal drawings, do you see a rat or a man’s face in the last image? The answer is likely a rat. Now watch the next round—after being shown pictures of faces, you might see a man’s face instead even though the final image hasn’t changed.

5. COLOR INFLUENCES TASTE ...

Every cooking show you’ve watched is right—presentation really is important. One look at something can dictate your expectations for how it should taste. Researchers have found that we perceive red food and drinks to taste sweeter and green food and drinks to taste less sweet regardless of chemical composition. Even the color of the cup we drink from can influence our perception of taste.

6. ... AND SO DOES SOUND

Sight isn’t the only sense that plays a part in how we taste. According to one study, listening to crunching noises while snacking on chips makes them taste fresher. Remember that trick before tossing out a bag of stale junk food.

7. BEING HYPER-FOCUSED HAS DRAWBACKS.

Have you ever been so focused on something that the world around you seemed to disappear? If you can’t recall the feeling, watch the video above. The instructions say to keep track of every time a ball is passed. If you’re totally absorbed, you may not notice anything peculiar, but watch it a second time without paying attention to anything in particular and you’ll see a person in a gorilla suit walk into the middle of the screen. The phenomenon that allows us to tune out big details like this is called selective attention. If you devote all your mental energy to one task, your brain puts up blinders that block out irrelevant information without you realizing it.

8. THINGS GET WEIRD WHEN SENSES CONTRADICT EACH OTHER.

Girl standing in optical illusion room.

The most mind-bending room in the "Our Senses" exhibit is practically empty. The illusion comes from the black grid pattern painted onto the white wall in such a way that straight planes appear to curve. The shapes tell our eyes we’re walking on uneven ground while our inner ear tells us the floor is stable. It’s like getting seasick in reverse: This conflicting sensory information can make us feel dizzy and even nauseous.

9. WE SEE SHADOWS THAT AREN’T THERE.

If our brains didn’t know how to adjust for lighting, we’d see every shadow as part of the object it falls on. But we can recognize that the half of a street that’s covered in shade isn’t actually darker in color than the half that sits in the sun. It’s a pretty useful adaptation—except when it’s hijacked for optical illusions. Look at the image above: The squares marked A and B are actually the same shade of gray. Because the pillar appears to cast a shadow over square B, our brain assumes it’s really lighter in color than what we’re shown.

10. WE SEE FACES EVERYWHERE.

The human brain is really good at recognizing human faces—so good it can make us see things that aren’t there. This is apparent in the Einstein hollow head illusion. When looking at the mold of Albert Einstein’s face straight on, the features appear to pop out rather than sink in. Our brain knows we’re looking at something similar to a human face, and it knows what human faces are shaped like, so it automatically corrects the image that it’s given.

All images courtesy of the American Museum of Natural History unless otherwise noted.

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