“Are you ready for a break from the road?” I asked Kathryn.

“Give me a few minutes’ warning so I can put on my shoes,” she said.

I drove for about ten more minutes and mulled over an item I had seen on the morning news about Donald Trump swiping at Hillary Clinton for a restroom break during a debate.

“Several gas stations here–Mobil, Love’s, Sheetz.  Let’s take a Sheetz…” I said.  And my scatological mind wandered over the array of applicable terms, at least one of which rhymed with the candidate’s last name.

Per our practice when driving solely to get from point A to point B, the rule is that at least two things must be taken care of at any stop. For instance, we can fuel up and go to the restroom. Or we can fuel up, go to the restroom, and get food. But we cannot just stop for a restroom break. And Kathryn does not like interstate rest areas, because there is basically only one thing you can do there. Service plazas, say along I-95, are OK though—restrooms, gas pumps (at a premium), and fast food abound.

She went inside the spacious, brightly-lighted Sheetz store while I opened the gas cap and inserted the pump nozzle. When she returned, I had finished topping off the car and went inside myself. Piped in country music blanketed the shoppers plying the rows of snack food and walls of soda. Kenny Chesney and Carrie Underwood are as popular in Pennsylvania as in Tennessee—maybe more. I followed the flow of males with urgent expressions on their faces, found the cavernous, clean restroom, and breathed a sigh of relief when I saw there was no queue. The urinal flushed itself when I stepped back, and the force of the water flicked a few droplets out on the floor. When I got to the sink, I reached to turn on the faucet, but there was no handle. I maneuvered my hands under the spigot for a couple of seconds to find the sweet spot where the sensor could detect my hands. A glance at the soap dispenser presented a moment’s confusion, because I could not tell whether you were supposed to push or pull to get an aliquot. I felt around on it, and then it emitted a low-high-low-high pitched servo-motor sound. A bit of clear blue hand soap went on the floor under the dispenser. The second time, I had my hand cupped under it. Perfume in the soap rose with the warm, moist column of air from the hot water, and I somehow knew it would cling to my hands for several minutes after leaving the building.

I looked around and found, to my delight, a paper towel dispenser. The electric dryers frustrate me, and I feel foolish holding my hands under them while a tepid breeze blows over my hands. Likely I’m wrong in thinking my hands get cleaner with a quick, efficient paper towel drying, but I’ll leave it for some enterprising high school students to do a science fair experiment to see if it really makes any difference. I waved my hands under the business end of the dispenser, and nothing happened, and then I noticed a little tail of paper sticking out. It was manual, so I grasped both sides with two hands and pulled down. Something resembling pinking shears inside scores or cut the sheet. In rapid succession, I pulled out three sheets, dried my hands on them, threw them in the trash can, and walked out.

Our practice, but not necessarily a rule, is to switch drivers at one of these stops, and Kathryn had already settled into the driver’s seat. I got in, and without her reminding me, buckled the seat belt.

“It was a 30,” I said, as Kathryn pulled out of the parking lot.


“The restroom.”

“What are you talking about?”

“You remember. In my model, there are thirty-two different types of public restrooms—I number them zero to thirty-one. Zero is totally manual, and thirty-one is totally automatic.”

“Oh, yeah. You and your math brain.”

“Can’t help that either.”

“How does it go again? Presumably thirty is pretty close to all-automatic.”

“Yes, in my scheme, 30 means everything is automatic except the towel dispenser. And 29 means everything except the soap dispenser is automatic.”

“Now, why are there 32 different kinds?”

“That’s where I have to idealize a little. Not every men’s room is configured quite the same way. For example, there might be a hand dryer, or a towel dispenser, or even both. I’m assuming there is just a paper towel dispenser. I think of it this way, when you go in the first thing you’re going to use is either the urinal or the toilet, and the former more often than the latter. So appliance number 1 is the urinal, followed by number 2, the toilet. After that, you’re probably going to wet your hands in the sink—appliance number 3—and then get some soap from appliance number 4. Once you’re done rinsing, you’re going to use appliance number 5, the paper towel dispenser.”


“Well, either the urinal is automatic or it isn’t. That’s two possibilities. After that, there are two possibilities for the toilet. Then, just looking at urinal and toilet together, there are two times two, or four, different urinal-toilet configurations—manual-manual, manual-auto, auto-manual, and auto-auto.”

“All right.”

“Bring the sink in. It’s either manual or automatic—two choices. So multiply the four by two to get eight. The same logic applies for the soap—two choices, so multiply eight by two to get sixteen. And finally, with the towel dispenser two more choices means multiply sixteen by two to get thirty-two.”

“Only you would think of something like that.”

“Now that I think about, … no. I beg to differ. Back in the eighties, Don Knuth published a paper in the American Mathematical Monthly called “The Toilet Paper Problem.”ToiletPaper

“I see. So you copied his idea.”

“No. I had not thought about anyone else’s bathroom math until you said that. My notion is lightweight stuff compared to Knuth’s paper. Amused by the title, I actually looked through it, and I distinctly remember him describing a scenario in which restroom stalls are supplied with two rolls of toilet paper, side by side—this was the old days before the slider thing that only allows access to one roll at a time—and that every user fit into one of two behavioral types, big-chooser or little-chooser. If the amount of TP on one roll was larger, then a big-chooser always chose from it, and a little-chooser always chose from the smaller one. If the amount of paper on the rolls was equal, it was a toss-up as to which one was chosen.”

“And why, exactly, is this guy pigeon-holing everyone this way?”

“It was to set up a concrete model for a mathematical system he was going to run and determine its long-term behavior.”

“Dare I ask, what’s the system?”

“The rest of the set-up was that he assumed everyone always used the same amount of paper, and that both rolls start out with the same number of ‘servings’ on them, say a hundred.  Also, he had one other assumption, that the proportions of big- and little-choosers were fixed in the population, say 2/3 big and 1/3 little. So he sets this system running, and at random people come in to use the toilet. About two-thirds of the time it’s a big-chooser, and one-third of the time it’s a little chooser. Keep this thing going until one TP roll is empty; then there’s still some left on the other roll. The question is, on average, how much TP is left on the other one?”

“You’re right, that sounds more complicated than your Baskin-Robbins style idea. Whew, could we talk about something else?”

“Don’t you want to know the punch line?”

“Well, I suppose you’re going to tell it to me whether I want to hear it or not.”

“It’s two.”


“Yes, if big-choosers predominate then on average when the first roll goes empty, the other one will have two portions on it.”

“What if it’s the other way around?”

“That’s the case that’s easier to intuit.  In that instance, it’s about fifty–assuming the roll started with a hundred portions.  You can see why.  If little-choosers predominate, then the smaller roll will get whittled down pretty quickly while the other one goes much more slowly.  On average, about half of the bigger roll disappears in the time it takes for the smaller roll to go away.”

In my peregrinations over the past eighteen months, I have been in countless public restrooms—in fast-food places, mom-n-pop gas stations, truck stops, bus stations, train stations, airports, rest-stops, workplace, restaurants, hotels, and office buildings. The sheer number of such structures is testimony to how many people there are in this world—in this country—and the variety of stages of automation communicates both the success of the marketers of these fancy devices and the wealth behind the organization that maintain the restroom. Some have been moderately nasty, and some have been astonishingly clean and industrialized. In the Atlanta airport—it was in an airport, I’m sure—there was an attendant who whistled and sang, found you an open stall, and bade you good day when you exited.

The occasional fumbling with automatic features of sinks and soap dispensers got me to thinking about why I should have any hesitation operating the equipment.  Part of the reason was that there are so many possible configurations. Thirty-two is several–not a huge number– and  enough so that if I’m in the habit of a thirty-one and then go into a twenty-three, then I might get off a rhythm. At least I had calculated a number that was reassuring: the plethora of possibilities helped explain why I just might have to pay attention in public restrooms.

Most folks think of mathematics as the way of telling what is. What’s the area of this room—I need to order carpet. What are the chances I win the jackpot on this Powerball ticket?  Is not wearing a black belt or black shoes the same thing as not wearing a black belt and not wearing black shoes?  Measure. Chance. Logic. Those are what math is about.

It is these things, but  math is also a big enough framework for engaging in what if.  What if the world consisted of big-choosers and little-choosers and everyone used the same amount of toilet paper in a public restroom? What if the number of babies born each year is the same fraction, year after year, of women between the ages of fifteen and forty-five? What if the universe is analogous to the surface of a doughnut, and when we look way out through telescopes toward some stars, are we actually looking at the back side of our own solar system?  What-if, practiced, becomes a habit of mind, becomes art. With art come aesthetic, and with aesthetic comes a deeper life.


Here’s a table that gives an explicit description of my restroom classification. Type zero is completely manual, and type thirty-one is totally automatic.

0 = fixture is manual

1 = fixture is automatic

Urinal Toilet Sink Soap Disp. Towel Disp. Restroom Type
0 0 0 0 0 0
0 0 0 0 1 1
0 0 0 1 0 2
0 0 0 1 1 3
0 0 1 0 0 4
0 0 1 0 1 5
0 0 1 1 0 6
. . . . . .
. . . . . .
. . . . . .
1 1 1 0 0 28
1 1 1 0 1 29
1 1 1 1 0 30
1 1 1 1 1 31





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