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.