Polly’s No-Meal is a popular new restaurant in the downtown Atlanta area. Though everyone enjoys their food, their patronage consists largely of one group in particular: mathematicians. It is believed that wild mathematicians are drawn to the restaurant by their name, which sounds similar to the word “polynomial.” Given their hit success with the mathematician community, Polly’s No-Meal has decided to cater their service directly to the tastes of mathematicians. They redesigned their menu in order to make it more mathematically oriented. All prices are given in duodecimal since the common mathematician has a tendency to emit a series of barks and growls, interspersed with theorems recited from memory, whenever confronted with a number that is not in the base numeral system of an antiprime. The entire establishment is also divided into two sides, pi and tau. This is because mathematicians have such strong preferences about these constants that if they were to be seated in the same area, chaos would ensue. All the tables in the establishment are equilateral triangles, allowing them to be tessellated together easily to form larger tables.
Due to these changes, Polly’s No-Meal has become a thriving restaurant, attracting various breeds of mathematicians from all over the city. Wild mathematicians have been frequently been seen covering their tables with spreadsheets and muttering to themselves that “No! My model is still off by 2.47%.” The napkins are used more often as scratch paper than as cleaning devices, and shouts of “Eureka!” can regularly be heard a quarter mile away.
Recently, the restaurant has encountered a bit of trouble. Last Monday, just as all the regular customers were slinking away to find some unaware English Major to prey on, a group of three mathematicians walked in with a menacing sway. As if on cue, all the light fixtures went out, leaving only the soft glow of the mathematicians’ ears. The mathematicians slowly approached Polly, with only soft wheezing noises coming from their eyes to permeate the silence. One, which wore a large fur coat covering a neon green swimsuit and a large hat that read “Elephants? More like baby terrorists,” spoke in a quiet, raspy voice. “What a nice little restaurant this is.” As he spoke, low clicking noises came from the general head regions of the other two mathematicians. “My associates and I enjoyed our meal, but were thoroughly unsatisfied with the dining experience itself. You see, mortal one, we ordered the alphabetic variable soup, and the container in which it was served did not have maximized volume efficiency. This…displeases us, and nobody wants to displease us.”
The first mathematician ceased speaking, and a second stepped forward. They wore a dark suit jacket over a darker black sweater-vest and a purple sequined fedora at an angle that enshrouded their face in darkness, making it difficult to discern their gender. They began to speak in that same gravelly voice, saying, “We will return in a week. We expect that by that time, you will have solved this little problem. If not, you shall face dire consequences. If we feel particularly merciful, we may just divide you by zero.” As they said this, the mathematician held up in one hand a crude effigy of Polly constructed out of wire, hay, and graph paper, and below it, in the other hand, a small zero. As they did this, the third mathematician, who wore a flowing black robe emblazoned with sacred geometry, stepped forward with a toothy smile showing her extra third row of teeth, a marked difference from the normal two rows. She held out a pointing finger, and made a sharp slashing motion. Simultaneously, a streak of flame shot in the same motion pattern between the doll and the zero, forming a fiery fraction bar. The effigy caught fire, and the mathematician in the fedora dropped it onto the floor and let it burn. The mathematician in the robe knelt down before the burning doll and began to chant in an obscure language. The chant roughly translated to, “We bring this offering to you, oh great Newton, master of our being, bringer of our prey, and owner of our souls. Just as you defeated Leibniz in unlocking the secrets of our universe, so too shall we defeat this mortal one, whose restaurant is an affront to our core belief in the sacred power of optimization. We shall rain destruction upon her in service to you, our great and powerful ruler. All praise is your rightful prize, oh great Newton.” After reciting this in about two minutes, the robed mathematician took the zero from the one in the fedora and dropped it onto the flaming effigy, causing the zero to vaporize instantly.
The mathematicians looked up from the doll. Simultaneously, they rasped, “Act wisely, mortal.” With that, the three mathematicians turned to leave. As they walked out, the robed one placed a small ring on one of the triangular tables. And then they left, with their double tails swaying behind them as they walked out of Polly’s No-Meal.
After the mathematicians left, Polly went to investigate the ring. It was a mobius loop, engraved with the words, “Fear the wrath of Newton,” around its singular face. She turned to face the doll, still burning on the tile floor. She retrieved the fire extinguisher and sprayed it at the doll, but the fire wouldn’t go out. After trying water and dark magic to remove the fire, she decided she would just have to wait it out. She stood vigil over that fire through the night. Then, as the clock struck 3:14 AM, the fire immediately went out, and the doll disintegrated into a pile of ash that smelled faintly of lavender and iron filings.
The next morning, as early mathematicians began lurking into the restaurant, Polly began working hastily on improving the design of her containers. Her soup containers have an important backstory. Back when Polly was first starting her business, she needed to buy land to build the restaurant on. Unfortunately, due to rising land prices and loan discrimination against people born at 285 feet or more above sea level, the only land she could afford was the site of a former combo chemical plant/Babies-R-Us. Due to the hazardous materials left over in the ground from the Babies-R-Us, like Cyanide, Nitroglycerin, and reverse chewing gum, the city would not legally allow her to open a restaurant there. After several weeks of searching law books and tomes of dark enchantments, Polly found just the legal loophole that she needed. If she could register her business as a packaging company, rather than a restaurant, she could legally operate on the site. She immediately began reconstructing her business model. The cheapest form of packaging she could sell was cardboard boxes, so she decided all her products would be overpriced cardboard boxes that came with a free meal inside each one. This way, she could legally sell her food while technically remaining a packaging company. To make it extra clear that her business was not a restaurant, she decided to name it Polly’s No-Meal to emphasize to authorities that she was not selling meals. After making those changes, the restaurant launched without a hitch, eventually growing to the thriving business it is today.
Polly sat down in her office/nuclear fallout shelter/guest room to work out just what to do about the mathematicians’ demands. She was no stranger to their shenanigans, so she knew she had better get down to it. The cardboard boxes she uses for her soups are folded from 25cm x 40cm sheets of cardboard. To fold them, she cuts out a square of the same size from each corner of the sheet, and then folds up the edges to form a sort of cardboard bowl. What size square should Polly cut out from each corner of the cardboard to maximize the volume contained by the bowl and appease the demands of the feral mathematicians?
5 cm