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.

11 Tips for Avoiding Germs at the Grocery Store

The early bird doesn't catch the germ.
The early bird doesn't catch the germ.
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While going to concerts, movie theaters, bars, beaches, and other recreational destinations is temporarily on hold, there’s one outing that remains a necessity during self-isolation: grocery shopping. If any supermarkets in your area offer home delivery or even store pickup, this is a good time to take advantage of those services.

But if you, like many of us, still need to stock up on food the old-fashioned way, here are some helpful tips for avoiding germs when you venture to the store.

1. Go early in the morning.

Not only will stores be less crowded in the early morning, but they’ll probably be cleanest then, too, since the staff often sanitizes the premises at night. Because many stores are devoting their early hours of operation to senior citizens only, Reader’s Digest suggests calling ahead to find out when your store opens to the general public.

2. Bring hand sanitizer, disinfectant wipes, or disposable gloves (and wipe down your cart).

Though many stores are now putting disinfectant wipes near the carts so you can wipe them down, you should bring your own just in case. This is especially important, since studies have found that COVID-19 can live for two or three days on plastic surfaces.

Your cart won’t be the only potentially germy place you put your hands during your trip—door handles in the frozen food section, self-checkout screens, and credit card keypads are all risky zones. Be sure to either wipe them down before touching, use hand sanitizer after touching, or just wear gloves that you can toss out at the end of your trip.

3. Don’t touch your face.

This may seem like a no-brainer, but you might be especially prone to absentmindedly touching your face while you contemplate which non-dairy milk to choose when your first choice is out of stock.

4. Don't touch your phone either.

Phone screens are a great example of high-touch surfaces where germs can live, so instead of keeping a grocery list on your smartphone, write it on a piece of scrap paper that you can throw away after you’re finished.

5. Give yourself more time to shop than you usually need.

Maintaining at least 6 feet between you and every other shopper means occasionally waiting for occupied aisles to clear and moving more slowly so you don’t run into people—not to mention the time it takes to use hand sanitizer or disinfectant wipes intermittently. If you’re trying to fit in a quick shopping trip before an important Zoom call with your boss, you may be less conscientious about shopping safely.

6. Inspect items for holes in the packaging (or the food itself).

Make sure there aren’t any rips or tears in cereal boxes, potato chip bags, or any other packaging—and that goes for produce, too. Give those apples a nice long look to be certain there aren’t any holes or breaks in the skin that germs could easily get into.

7. Bypass the free samples.

Surprise snacks at supermarkets are one of the perks of grocery shopping, but Livestrong points out that exposed food is an easy target for germs. So skip the free samples and don't graze on those bunches of grapes; instead, reward yourself with an extra snack at home. Some stores, like Costco, are even suspending their samples during this time, so you won't be so tempted.

8. Don’t pay with cash.

While there’s a certain satisfaction in counting out exact change, cash has a reputation for being a hotbed for germs. If possible, stick to cards or other automatic methods of payment. Even then, it's not the worst idea in the world to wipe down debit and credit cards after using them.

9. Leave the grocery bags on your doorstep.

Store employees are being extra cautious about cleanliness, but it’s still possible that your bags could pick up germs during the checkout process. To avoid the risk, leave them outside and only bring your items into the house.

10. Wash reusable bags between trips.

If you’ve made the switch to reusable shopping bags, Food Network recommends tossing them in the washing machine or wiping them down with soap and water between shopping trips.

11. Wash produce and wipe down other items.

Per usual, you should thoroughly rinse produce before eating it. Dr. Lisa Larkin, a Cincinnati-based internal medicine physician and founder of Ms.Medicine, told Reader’s Digest that you can also wipe down jars, cans, and bottles with a disinfectant wipe before putting them in your pantry for good measure.

[h/t Reader’s Digest]

12 Things You Might Not Know About Passover

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For practicing Jews, Passover is a time to remember their deliverance from captivity in ancient Egypt. It's one of the most important holidays on the Jewish calendar, and in the days before the first night's seder, families make preparations such as cleaning the home of chametz and planning for a week of meaningful dietary restrictions. Here are 12 facts about Passover that you wouldn't have learned from a yearly viewing of The Ten Commandments.

1. Firstborn sons need to fast for Passover.

matzo
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The festival of Passover (or Pesach) commemorates the story of the Jews' escape from Egypt. The passover in question is when the houses of the observant Israelites in captivity were "passed over" as Egypt's first-born children were killed (although confusingly, in the Torah, the date the 14th of Nisan is referred to as Passover while the week-long celebration is the Festival of Matzot. They've since been combined into one celebration called Passover).

In celebration of the firstborns being saved, it is traditional for them to fast on 14 Nisan. If there are no children, the oldest member of the household fasts. If the firstborn is a daughter? That depends on the tradition of the community.

2. Passover lasts either seven or eight days.

reading the Haggadah at Passover
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The Torah says to celebrate Passover for seven days (the time between the Exodus and the parting of the Red Sea), but many Jews outside of Israel celebrate for eight. Traditionally each month of the Jewish calendar was determined by an astronomical observation and could be either 29 or 30 days long. After a new month was determined, messengers spread the word. For Jews who lived too far away for messengers to bring timely news of a new month, it was safest to celebrate for an extra day, so no matter how long the previous month was, the holiday was celebrated.

Eventually the calendar was standardized and the eight-day custom was no longer needed. Today, some Jewish denominations outside of Israel (like Reform Judaism) celebrate the mandated seven days, while many others prefer eight days. Inside Israel it's generally seven.

3. Leavened grains are a no-go at Passover.

Person sweeping the floor
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One of the most important parts of Passover preparations is cleaning the house of chametz, or leavened food. Even the tiniest bit has to go. Because the Jews left Egypt in such a hurry, it's said they didn't have time to leaven their bread. To commemorate that, five grains (traditionally wheat, barley, rye, spelt, and oats) are banished from the house. Jews can spend weeks ensuring that the house is perfectly clean—and there are even professional chametz cleaning services that say they'll boil toys, break down and reassemble kitchen chairs … and possibly still leave the house dirty. There's a saying in Jewish households: "Dust is not chametz." The goal is to get rid of chametz above all else.

4. Matzo, which is made from wheat, is one of the most important parts of a Passover meal.

baking matzo
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While there are restrictions against leavened products, one of the most important parts of a Passover meal is matzo, which is made from wheat. The difference between matzo and regular bread is that the wheat in matzo cannot come into contact with any water until it's ready to be cooked. And once water and wheat are mixed it has to be baked within 18 minutes (sources differ as to whether the timer stops when it enters or leaves the oven). After 18 minutes, fermentation begins and it is chametz.

But why 18 minutes? Supposedly it's because that's how long it takes to walk between the cities of Migdal Nunaiya and Tiberias in Israel. Over the years, scholars have argued about how long it would actually take to walk between the cities, with some proposing that copying errors reduced the distance from circa 4 miles to 1 and thus reduced the time from 72 minutes to 18. Nowadays, it's felt that even if there was a transcribing error, there's enough tradition to use 18 minutes.

5. Grains get complicated during Passover.

matzo ball soup
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As Jews spread around the world, they often found themselves faced with foods that weren't explicitly mentioned for Passover. Sephardic Jews (generally) feel that only the five expressly mentioned grains are forbidden, while Ashkenazi Jews worry that the dishes made from certain other plants that look similar and are grown in similar conditions as the forbidden grains will risk contamination between the two. So if these ingredients (called kitniyot, or "legumes") were avoided, actual chametz could more easily be avoided (although kitniyot is nowhere near as regulated as chametz).

But recently, some authorities have argued that improved technology and storing methods have rendered the old methods obsolete. It's a current debate in some communities.

6. Some of the best matzo flour is made in Arizona.

field of wheat
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One of the most difficult parts of making matzo is keeping the flour dry before it's ready to be converted into matzo; any water risks converting flour into chametz. So, according to The New York Times, one sect of Hasidic Jews has found the perfect farming conditions to produce their wheat—the arid fields of southwestern Arizona. The group of ultra-Orthodox Jews from Brooklyn, New York, work with a farm in Yuma, Arizona, to ensure that no unwanted moisture affects the crop, and the resulting kosher wheat is shipped back east to make up to 100,000 pounds of matzo.

7. Pets also get special food during Passover.

cute dog with head tilted
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For Passover, houses must be free of chametz and there can be no benefit derived from it. This includes pet food. In keeping with this, there are Passover-friendly pet foods out there, and some Rabbinical authorities propose switching out your pet's diet for a few days—such as giving dogs straight meat or herbivores a variety of approved vegetables. If a pet must have a specific type of food—or you can't get Passover-friendly pet food—some observant Jews follow the rabbinical authorities who give the option to sell the pet to a gentile for a few days and then get it back after Passover has ended.

8. There are six symbolic Passover foods.

seder plate for Passover
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The focal point of the start of Passover is the Seder plate, and on it are six ceremonial items:

Beitzah—A cooked egg, representing sacrifice (it's also been suggested that while most foods soften when you cook them, eggs get harder, representing the resolve of the Jewish people)

Haroset—a sweet mix of fruits, nuts, and honey/wine that symbolizes the mortar used by Jews during their slavery

Karpas—a green vegetable signifying new life

Maror and hazeret—bitter herbs (often horseradish for maror and something like romaine for hazeret) to represent the bitterness of slavery

Zeroa—a shank bone (or a chicken neck) to remember the Paschal sacrifice.

9. Sometimes an orange is added to the Seder plate.

slice of orange
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In the 1980s, Dartmouth professor Susannah Heschel spoke on a panel at Oberlin College. While there, she met some students who told a story of a rabbi who said "There's as much room for a lesbian in Judaism as there is for a crust of bread on the seder plate." In response, they started placing a crust on their plates.

Heschel was inspired, but felt that using bread sent the wrong message, writing "it renders everything chametz … [suggesting] that being a lesbian is being transgressive, violating Judaism." So she proposed putting an orange (originally a tangerine) on the Seder plate to symbolize Jewish gays and lesbians. At some point a story emerged that it was actually to symbolize women in general, but Heschel explained: "A woman's words are attributed to a man, and the affirmation of lesbians and gay men is erased. Isn't that precisely what's happened over the centuries to women's ideas?"

Other more modern additions include pine cones (symbolizing mass incarceration), an artichoke (to recognize interfaith families), or tomatoes or Fair Trade chocolate (to remember that there's still slavery around the world).

10. Some major companies produce special kosher-for-Passover food and beverages.

ad for kosher Coca-Cola
Mike Mozart, Flickr // CC-BY-2.0

Many companies produce special kosher-for-Passover products, from chocolate syrup to cake mixes. But one of the most important is Coca-Cola. In the early 20th century Rabbi Tobias Geffen was serving as an Orthodox Rabbi in Atlanta. Due to his location (Coca-Cola was invented and is headquartered in Atlanta), he was frequently asked if Coca-Cola was kosher. After analyzing the product, he found two problem ingredients—alcohol and glycerin.

The alcohol was a problem because it was grain-derived and thus unacceptable for Passover, a problem that was solved by switching to fermented molasses. The other problem, however, was glycerin. The glycerin was derived from animals, and there was simply no economic way to ensure the animals were kosher. As Roger Horowitz explains in Kosher USA, there's an exemption in the rules for a tiny amount of an unacceptable ingredient—designed to cover mistakes—and Coca-Cola's glycerin content was dramatically below that level. Rabbi Geffen, however, believed that since the glycerin was deliberately added, it didn't qualify for this rule. Soon though, a new source of glycerin from cottonseed oil emerged, and Coca-Cola was approved for Passover.

When Coca-Cola switched to high fructose corn syrup, however, that created a problem for Ashkenazi Jews. As such, today there's a special yellow-capped Coca-Cola that doesn't use HFCS and is certified kosher.

11. Maxwell House coffee holds a special place at Passover.

Maxwell House Haggadahs
Tom Lappin, Flickr // CC BY-NC 2.0

But the most influential company is likely Maxwell House. In the 1920s they decided to expand their presence to Jewish families—but there was a problem. Colloquially known as coffee "beans," there was a view that they were legumes, and as such forbidden to Ashkenazi Jews. Soon Maxwell House convinced reluctant coffee drinkers that their product was acceptable and in 1932 the company began publishing the Maxwell House Haggadah (the Haggadah is the telling of the Exodus and how to perform a seder meal). In the years since, Maxwell House estimates that it has published 50 million Haggadahs, which were even the preferred text for the Obama White House Seder.

12. The world's largest Seder happens in a surprising location.

Hundreds of worshippers gather in a hall for Passover in Kathmandu in 2014.
Hundreds of worshippers gather in a hall for Passover in Kathmandu in 2014.
PRAKASH MATHEMA, AFP/Getty Images

Going on for almost 30 years and hosting over 1000 people, the Kathmandu Seder was started in 1989 by the Israeli ambassador to Nepal, who quickly realized that the demand was much higher than he was ready for. The ambassador contacted a rabbi friend who dispatched two rabbinical students to aid the preparations. The seder was a massive success—expecting 90 guests and hoping for 150, they ultimately had 500 guests.

Nowadays, preparations for the seder start months in advance, with 1000 bottles of wine and over 1000 pounds of matzo getting shipped in from the United States and Israel.

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