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

iStock
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.

Amazon's Best Black Friday Deals: Tech, Video Games, Kitchen Appliances, Clothing, and More

Amazon
Amazon

This article contains affiliate links to products selected by our editors. Mental Floss may receive a commission for purchases made through these links.

Black Friday is finally here, and Amazon is offering great deals on kitchen appliances, tech, video games, and plenty more. We will keep updating this page as sales come in, but for now, here are the best Amazon Black Friday sales to check out.

Kitchen

Instant Pot/Amazon

- Instant Pot Duo Plus 9-in-115 Quart Electric Pressure Cooker; $90 (save $40)

- Keurig K-Cafe Special Edition; $190 (save $30)

- Ninja OS301 Foodi 10-in-1 Pressure Cooker and Air Fryer; $125 (save $75)

- Nespresso Vertuo Next Coffee and Espresso Machine by Breville; $120 (save $60)

- KitchenAid KSMSFTA Sifter with Scale Attachment; $95 (save $75)

- Keurig K-Mini Coffee Maker; $60 (save $20)

- Cuisinart Bread Maker; $80 (save $97)

- Anova Culinary Sous Vide Precision Cooker; $139 (save $60)

- Aicook Juicer Machine; $35 (save $15)

- JoyJolt Double Wall Insulated Espresso Mugs - Set of Two; $14 (save $10)

- Longzon Silicone Stretch Lids - Set of 14; $16 (save $11)

- HadinEEon Milk Frother; $37 (save $33)

Home Appliances

Roomba/Amazon

- iRobot Roomba 675 Robot Vacuum with Wi-Fi Connectivity; $179 (save $101)

- ASAKUKI 500ml Premium Essential Oil Diffuser; $22 (save $4)

- Facebook Portal Smart Video Calling 10 inch Touch Screen Display with Alexa; $129 (save $50)

- Bissell air320 Smart Air Purifier with HEPA and Carbon Filters; $280 (save $50)

- Oscillating Quiet Cooling Fan Tower; $59 (save $31)

- TaoTronics PTC 1500W Fast Quiet Heating Ceramic Tower; $55 (save $10)

- Vitamix 068051 FoodCycler 2 Liter Capacity; $300 (save $100)

- Ring Video Doorbell; $70 (save $30)

Video games

Sony

- Marvel's Spider-Man: Game of The Year Edition for PlayStation 4; $20 (save $20)

- The Legend of Zelda: Link's Awakening; $40 (save $20)

- Hyrule Warriors: Age of Calamity; $50 (save $10)

- Marvel's Avengers; $25 (save $33)

- The Last of Us Part II for PlayStation 4; $30 (save $30)

- LEGO Harry Potter: Collection; $15 (save $15)

- Ghost of Tsushima; $40 (save $20)

- BioShock: The Collection; $20 (save $30)

- The Sims 4; $24 (save $20)

- God of Warfor PlayStation 4; $10 (save $10)

- Days Gonefor PlayStation 4; $20 (save $6)

- Luigi's Mansion 3 for Nintendo Switch; $40 (save $20)

Computers and tablets

Microsoft/Amazon

- New Apple MacBook Pro 16 inches with 512 GB; $2149 (save $250)

- Microsoft Surface Laptop 3 with 13.5 inch Touch-Screen; $1200 (save $400)

- Lenovo ThinkPad T490 Laptop; $889 (save $111)

- Amazon Fire HD 10 Tablet (64GB); $120 (save $70)

- Amazon Fire HD 10 Kids Edition Tablet (32 GB); $130 (save $70)

- Apple iPad Mini (64 GB); $335 (save $64)

- Vankyo MatrixPad S2 Tablet; $120 (save $10)

Tech, gadgets, and TVs

Apple/Amazon

- Apple Watch Series 3 with GPS; $120 (save $79)

- Seneo Wireless Charger, 3 in 1 Wireless Charging Station; $16 (save $10)

- SAMSUNG 75-inch Class Crystal 4K Smart TV; $998 (save $200)

- Nixplay 2K Smart Digital Picture Frame 9.7 Inch Silver; $238 (save $92)

- All-New Amazon Echo Dot with Clock and Alexa (4th Gen); $39 (save $21)

- MACTREM LED Ring Light 6" with Tripod Stand; $16 (save $3)

- Amazon Fire TV Stick with Alexa Voice Remote; $28 (save $12)

- DR. J Professional HI-04 Mini Projector; $93 (save $37)

Headphones and speakers

Beats/Amazon

- Beats Solo3 Wireless On-Ear Headphones; $120 (Save $80)

- Apple AirPods Pro; $169 (save $50)

- Anker Soundcore Upgraded Bluetooth Speaker; $22 (save $8)

- Powerbeats Pro Wireless Earphones; $175 (save $75)

- JBL Boombox; $280 (save $120)

Movies and TV

HBO/Amazon

- Game of Thrones: The Complete Series; $115 (save $89)

- Jurassic World 5-Movie Set; $23 (save $37)

- Deadwood: The Complete Series; $42 (save $28)

- Back to the Future Trilogy; $15 (save $21)

Toys and Games

Amazon

- Awkward Family Photos Greatest Hits; $15 (save $10)

- Exploding Kittens Card Game; $10 (save $10)

- Cards Against Humanity: Hidden Gems Bundle; $14 (save $5)

- LOL Surprise OMG Remix Pop B.B. Fashion Doll; $29 (save $6)

- LEGO Ideas Ship in a Bottle 92177 Expert Building Kit; $56 (save $14)

Furniture

Casper/Amazon

- Casper Sleep Element Queen Mattress; $476 (save $119)

- ZINUS Alexis Deluxe Wood Platform Bed Frame; $135 (save $24)

- ROMOON Dresser Organizer with 5 Drawers; $59 (save $11) 

- AmazonBasics Room Darkening Blackout Window Curtains; $26 (save $5)

- Writing Desk by Caffoz; $119 (save $21)

- SPACE Seating Office Support Managers Chair; $112 (save $116)

- Rivet Globe Stick Table Lamp; $53 (save $17)

- Christopher Knight Home Merel Mid-Century Modern Club Chair; $188 (save $10)

- Walker Edison Furniture Industrial Rectangular Coffee Table; $121 (save $48)

Beauty

Haus/Amazon

- MySmile Teeth Whitening Kit with LED Light; $21 (save $12) 

- Cliganic USDA Organic Lip Balms Set of Six; $6 (save $4)

- HAUS LABORATORIES By Lady Gaga: LE RIOT LIP GLOSS; $7 (save $11)

- Native Deodorant for Men and Women Set of Three; $25 (save $11) 

- BAIMEI Rose Quartz Jade Roller & Gua Sha; $14 (save $3)

- Honest Beauty Clearing Night Serum with Pure Retinol and Salicylic Acid; $20 (save $8)

- WOW Apple Cider Vinegar Shampoo and Hair Conditioner Set; $30 (save $5) 

- La Roche-Posay Effaclar Purifying Foaming Gel Cleanser; $15 (save $5)

- wet n wild Bretman Rock Shadow Palette; $9 (save $6)

- EltaMD UV Daily Tinted Face Sunscreen Moisturizer with Hyaluronic Acid; $25 (save $6)

Clothes

Ganni/Amazon

- Ganni Women's Crispy Jacquard Dress; $200 (save $86) 

- The Drop Women's Maya Silky Slip Skirt; $36 (save $9)

- Steve Madden Women's Editor Boot; $80 (save $30)

- adidas Women's Roguera Cross Trainer; $40 (save $25)

- Line & Dot Women's Elizabeth Sweater; $74 (save $18)

- Levi's Men's Sherpa Trucker Jacket; $57 (save $41)

- Adidas Men's Essentials 3-Stripes Tapered Training Joggers Sweatpants; $28 (save $12)

- Timex Men's Weekender XL 43mm Watch; $32 (save $20)

- Ray-Ban Unisex-Adult Hexagonal Flat Lenses Sunglasses; $108 (save $46) 

- Reebok Men's Flashfilm Train Cross Trainer; $64 (save $16)

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10 Little Facts About Louisa May Alcott

Public Domain, Wikimedia Commons
Public Domain, Wikimedia Commons

Born on November 29, 1832, Louisa May Alcott led a fascinating life. Besides enchanting millions of readers with her novel Little Women, she worked as a Civil War nurse, fought against slavery, and registered women to vote. Here are 10 facts about the celebrated author.

1. Louisa May Alcott had many famous friends.

Louisa's parents, Bronson and Abigail Alcott, raised their four daughters in a politically active household in Massachusetts. As a child, Alcott briefly lived with her family in a failed Transcendentalist commune, helped her parents hide slaves who had escaped via the Underground Railroad, and had discussions about women’s rights with Margaret Fuller.

Throughout her life, she socialized with her father’s friends, including Henry David Thoreau, Ralph Waldo Emerson, and Nathaniel Hawthorne. Although her family was always poor, Alcott had access to valuable learning experiences. She read books in Emerson’s library and learned about botany at Walden Pond with Thoreau, later writing a poem called "Thoreau’s Flute" for her friend. She also socialized with abolitionist Frederick Douglass and women’s suffrage activist Julia Ward Howe.

2. Louisa May Alcott's first nom de plume was Flora Fairfield.

As a teenager, Alcott worked a variety of teaching and servant jobs to earn money for her family. She first became a published writer at 19 years old, when a women’s magazine printed one of her poems. For reasons that are unclear, Alcott used a pen name—Flora Fairfield—rather than her real name, perhaps because she felt that she was still developing as a writer. But in 1854 at age 22, Alcott used her own name for the first time. She published Flower Fables, a collection of fairy tales she had written six years earlier for Emerson’s daughter, Ellen.

3. Louisa May Alcott secretly wrote pulp fiction.

Before writing Little Women, Alcott wrote Gothic pulp fiction under the nom de plume A.M. Barnard. Continuing her amusing penchant for alliteration, she wrote books and plays called Perilous Play and Pauline’s Passion and Punishment to make easy money. These sensational, melodramatic works are strikingly different than the more wholesome, righteous vibe she captured in Little Women, and she didn’t advertise her former writing as her own after Little Women became popular.

4. Louisa May Alcott wrote about her experience as a Civil War nurse.

In 1861, at the beginning of the U.S. Civil War, Alcott sewed Union uniforms in Concord and, the next year, enlisted as an army nurse. In a Washington, D.C. hotel-turned-hospital, she comforted dying soldiers and helped doctors perform amputations. During this time, she wrote about her experiences in her journal and in letters to her family. In 1863, she published Hospital Sketches, a fictionalized account, based on her letters, of her stressful yet meaningful experiences as a wartime nurse. The book became massively popular and was reprinted in 1869 with more material.

5. Louisa May Alcott suffered from mercury poisoning.

After a month and a half of nursing in D.C., Alcott caught typhoid fever and pneumonia. She received the standard treatment at the time—a toxic mercury compound called calomel. (Calomel was used in medicines through the 19th century.) Because of this exposure to mercury, Alcott suffered from symptoms of mercury poisoning for the rest of her life. She had a weakened immune system, vertigo, and had episodes of hallucinations. To combat the pain caused by the mercury poisoning (as well as a possible autoimmune disorder, such as lupus, that could have been triggered by it), she took opium. Alcott died of a stroke in 1888, at 55 years old.

6. Louisa May Alcott wrote Little Women to help her father.

In 1867, Thomas Niles, an editor at a publishing house, asked Alcott if she wanted to write a novel for girls. Although she tried to get excited about the project, she thought she wouldn’t have much to write about girls because she was a tomboy. The next year, Alcott’s father was trying to convince Niles to publish his manuscript about philosophy. He told Niles that his daughter could write a book of fairy stories, but Niles still wanted a novel about girls. Niles told Alcott’s father that if he could get his daughter to write a (non-fairy) novel for girls, he would publish his philosophy manuscript. So to make her father happy and help his writing career, Alcott wrote about her adolescence growing up with her three sisters. Published in September 1868, the first part of Little Women was a huge success. The second part was published in 1869, and Alcott went on to write sequels such as Little Men (1871) and Jo’s Boys (1886).

7. Louisa May Alcott was an early suffragette.

In the 1870s, Alcott wrote for a women’s rights periodical and went door-to-door in Massachusetts to encourage women to vote. In 1879, the state passed a law that would allow women to vote in local elections on anything involving education and children—Alcott registered immediately, becoming the first woman registered in Concord to vote. Although met with resistance, she, along with 19 other women, cast ballots in an 1880 town meeting. The Nineteenth Amendment was finally ratified in 1920, decades after Alcott died.

8. Louisa May Alcott pretended to be her own servant to trick her fans.

After the success of Little Women, fans who connected with the book traveled to Concord to see where Alcott grew up. One month, Alcott had a hundred strangers knock on the door of Orchard House, her family’s home, hoping to see her. Because she didn’t like the attention, she sometimes pretended to be a servant when she answered the front door, hoping to trick fans into leaving.

9. Louisa May Alcott never had children, but she cared for her niece.

Although Alcott never married or had biological children, she took care of her orphaned niece. In 1879, Alcott’s youngest sister May died a month after giving birth to her daughter. As she was dying, May told her husband to send the baby, whom she had named Louisa in honor of Alcott, to her older sister. Nicknamed Lulu, the girl spent her childhood with Alcott, who wrote her stories and seemed a good fit for her high-spiritedness. Lulu was just 8 when Alcott died, at which point she went to live with her father in Switzerland.

10. Fans can visit Louisa May Alcott's home in Concord, Massachusetts.

At 399 Lexington Road in Concord, Massachusetts, tourists can visit Orchard House, the Alcott family home from 1858 to 1877. Orchard House is a designated National Historic Landmark, and visitors can take a guided tour to see where Alcott wrote and set Little Women . Visitors can also get a look at Alcott’s writing desk and the family’s original furniture and paintings.