The Laziness Index

By Bobby Neal Winters

I got a call the other day from my old friend Bubba back home.  He was being strangely thoughtful.

I often forget that Bubba received a college education because, I suppose, he so rarely gives any evidence of it.  The fact the of matter is that Bubba could always be quite successful at anything he wanted to do. He never suffered from a lack of IQ; it was more a deficit of “want-to.”

“So,” he began slowly, “you’ve heard of the heat index, haven’t you?”

“Yes,” I replied.  I’d heard a lot about it recently as it had been over 100 degrees and the heat index hand been exceeding that level of five or ten degrees. “We’ve been experiencing it directly.

“Well,” again he was approaching the subject with uncharacteristic care and thoughtfulness, “is it real or just something made up?”

“What do you mean?” I asked.  I am often not very sure where he is coming from, and, though I had an idea, there are times when it is just better to play dumb and ask questions.

“I guess I am asking if it corresponds to reality as described by modern physics or if is just something invented by the pretty talking heads they hire to report on the weather so they can get people to tune in and learn how miserable they are?”  Bubba can be pretty blunt.

“First of all,” I said, “not all of the folks who talk about the weather are all that pretty, and second some are pretty smart.  But to answer your question it is an attempt to capture something real.”

I went on to talk about the affect of humidity.  Humidity can have a tremendous effect on comfort. The human animal uses perspiration as a natural means of air-conditioning.  The sweat glands release water, i.e. sweat. 

When this water evaporates, it absorbs heat and that heat is moved away from the body.  In areas of high relative humidity, the rate of evaporation is slowed because there is a limit to how much moisture air can hold at any given temperature.

“There’s a formula for it, isn’t there?” he asked. “Do you know what it is?”

“There is a formula for it,” I said. “But I don’t know if off of the top of my head.”

“I thought you were a math teacher,” he said, sounding superior now. “What do you folks do if it’s not remembering formulas?”

He does this just to hector me.  In the past, I have attempted to justify my existence both as a mathematician and a teacher of mathematics, but I’ve never gotten anything to stick.  I’ve finally decided that he does this simply to annoy me, so, to avoid rewarding this behavior, I just ignore him.

“We do other things,” I said. “Besides it’s something I can look up anytime on the Internet.”

“What about wind chill?” he asked.  “Is there a formula for that too?”

“Yes,” I said, and, seeing where this was going, I added, “and I don’t know that one either because I can look it up any time.”

In my mind, I was being subtle.  I was attempting to plant the suggestion that, as he spends hours upon hours on the Internet surfing from Sasquatch site to Sasquatch site and finding the perfect fishing lure, he might be able to look this up himself.  I so easily forget that such methods are futile when dealing with Bubba.

“I’m glad to hear that you can look them up any time,” he said, “because I want you to look them up for me and explain them to me.”

I could see that I’d been out-maneuvered, so when we finished our conversation, I went to the Internet and looked the formulas up.

It turns out that, while the formulas are not sophisticated from the point of view of a mathematician, they are complicated from the point of view of the man-on-the-street--or the Bubba-on-the-dirt-road if you prefer. The formula for the heat index can be stated as follows:

                                       

The variable T stands for temperature measured in degrees Fahrenheit and R stands for relative humidity as a percent, i.e. 50 rather than 0.50.  The subscripted “a” variables stand in for particular values that I will now give below:

                                                   

This formula involves no exotic functions, such as logarithms or the various trigonometric functions; it only requires addition, subtraction, and multiplication.  But it contains nine terms and the coefficients--the “a” variables--are decimals and some require scientific notation to represent.  In doing actual calculations with this formula, one would be well-advised to arrange his work carefully.  I myself prefer to put this sort of calculation in a spreadsheet.

The formula for wind chill is shorter, but it is a tad more exotic:

                                        

Again the T is temperature in degrees Fahrenheit while the V is wind speed in miles per hour.  The coefficients--the “b” variables--are decimals as given below:

                                                         

What makes it a little more sophisticated that the formula for heat index is that the V variable sports an exponent of 0.16.  This requires a special key on the calculator to evaluate, so it is no surprise that a spreadsheet is the best place to implement this calculation.

After looking these up and playing with them in a spreadsheet, I did a little report on it for Bubba and sent it off in an email. 

I was also careful to add that the heat index formula was only good for temperatures higher than 80 and the wind chill formula was only good for temperatures lower than 50. I think that may have been a mistake on my part, because it seemed to have turned Bubba’s thoughts in a particular direction.

My cell phone rang and the called ID read “Bubba.”  I took a deep breath, as has become my practice, and answered.

“Hello.”

“Those are some durned old complicated formulas there,” he said. “Can’t you come up with anything simpler?”

“They are what they are,” I replied.

“You know,” he was now getting thoughtful again, “The fact that the heat index only works above 80 degrees and the wind chill index only works below 50 degree got me to thinking that these formulas are just ways of telling you when it’s too bad to work outside.”

“That’s not a bad way to think about it,” I said.  Being a teacher, I always try to encourage thinking that is in the right direction.

“That made me wonder whether we might turn that around a little bit,” he said.

I was beginning to get nervous.

“Oh?” was all I could manage.

“Yeah,” he replied.  “I got to wondering whether maybe we out to figure out when it’s to good to work outside.”

“To good to work outside?” I was honestly confused.  “What do you mean?”

“Well, now, if it’s 70 degrees outside with a light breeze and there are a few clouds here and that and the fish are jumping, then I don’t want to waste a day like that on work.  I was wondering if you might could come up with a formula to say that.”

“What?”

“Yeah, you could factor in the temperature, the percentage cloud cover, whether the fish were biting, the price of beer...”

I cut him off.

“You know what?  I think the weather might be too good even now for me to work on that formula.”

“Maybe I could’ve figured that,” he said.

And we hung up.

Foreign, sick science

By Bobby Neal Winters

As I stood looking at the body on the table, waiting for Suzanna Doughcoup, our local homicide detective, to arrive, I began to examine my own character more closely.

Why do I find it so hard to say no when someone asks me for a favor?

Why is this especially difficult when it has something to do with mathematics?

Why do I even consider these things when it is something so far out of my area of expertise?

I could find no answers, at least none that I wanted to accept, but I will have to admit I don’t do my best thinking within the cold, antiseptic, rather creepy confines of a morgue. 

The call had come at 4am.  I staggered to the phone which is placed some distance from my bed.  As badly as I hate getting woken up by the phone, it is much worse when it is right by my head.

I don’t remember saying hello but I must’ve because the voice on the other end of the phone spoke to me.

“Hello,” it said.  It seemed to be vaguely feminine.

“What?” I replied.

“Hello,” the voice came back and called me by name.  “This is Detective Suzanna Doughcoup.”

“Who?” I asked.  I still wasn’t taking this in.

“Detective Suzanna Doughcoup,” she said. “I need your help to determine a time of death.”

“Death?” I was really doing rather badly. “Time?  What time is it?”

“It’s 4am,” Detective Doughcoup answered. “But that’s not important. What I need is for you to help me with a calculation of a time of death.  Will you do it?”

Four in the morning is really not a good time for me.  I get in some of my heaviest REM sleep in along about them.

“Do it?” I asked.  My inflection must have been off a little because I don’t think Suzanna heard the question mark on the end of that.

“Great!” she said.  “I will either come to get you or send a car.  That will take about half an hour.”

She hung up the phone.

I looked back at the bed which was beckoning to me.  It was singing whatever song Circe sang to Odysseus.  It was singing it quite well too.  I almost yielded to it, but I had a vision of Detective Suzanna Doughcoup battering down my front door and dragging me to the police station in my underwear. 

I pulled on my blue jeans, a t-shirt, and my tennis shoes.  Then I went to the kitchen and made a pot of coffee.  The coffee was done and I’d filled a thermos with it by the time I saw the squad car roll up.

It took the coffee time to perk, and, during that time, I got on the Internet to look up information on time of death calculations. During that brief interval of time, I was able to find five different ways of estimating the time of death: pallor mortis, livor mortis, algor mortis, rigor mortis, and decomposition.

 Pallor mortis is paleness and it begins fifteen minutes after death and lasts until two hours after death.  Livor mortis is a settling of the blood in capillaries in a way that causes purplish discoloration; maximum discoloration is 6 to 12 hours after death. Rigor mortis starts approximately three hours after death and lasts for approximately three days. I couldn’t even make myself look up the details on decomposition; Detective Doughcoup would strictly be on her own if decomposition were involved.

The remaining one, algor mortis, interested me the most.  Algor mortis is the cooling the body that follows death.  Using a mathematical formula, one can estimate the time of death.  I’d seen it on all my favorite detective shows, but I’d never had the occasion to look it up.

I thought that I knew how it was done. When I’d taught differential equations a number of years ago, I came upon Newton’s Law of Cooling, named for none other than Sir Isaac Newton.  As the story goes, an apple hit him on the head and he discovered Newton’s Law of Gravity.  I can only imagine that someone threw some cold water on him and he discovered the Law of Cooling.

In plain language, it states that the colder the environment is than an object, the quicker the object cools off.  That sounds like common sense and it is common sense, but when you translate it into mathematics it looks more mysterious:

Here the thing on the left hand side of the equation that looks like a fraction is the rate of change of temperature with respect to time. 

On the right hand side, the h is a constant that depends on the physical substance being cooled. Metal cools quicker than wood, for example.

The A is the surface area that is exposed.  Something that exposes more surface area will cool more quickly than something with less surface area.

The

on the right hand side, represents the difference between the environmental temperature and the temperature of the object.  The colder the environment is, the quicker the object cooler.

Quite frankly, the possibility of applying this equation was making my mouth water.  The solution involved logarithms, and logarithms are cool. 

I will admit that logarithms get a bad rap in our popular culture.  In the movie An Officer and a Gentleman, one of the characters refers to them with a participle that begins with the letter “f.” In Roughing It, Mark Twain calls one of his not-too-bright traveling companions a logarithm.

I’ve been of the opinion that giving them the name “logarithms” was a huge marketing mistake.  They should’ve called them “happy numbers” or something, but I digress.

In any case, I’d gotten excited about the prospect of using logarithms and expected to see them in the formula used to calculate time of death.  Imagine my surprise when I find Newton’s Law of Cooling alluded to only in passing and am presented with the Glaisters Equation instead. 

The Glaisters Equation is:

On the left hand side, the little t is the time since death.  On the right hand side, the T_R  is the rectal temperature of the cadaver.  I suppose it is hard to get them to put the thermometer under their tongue.

The taking of the temperature is the hardest part of this formula.  You subtract the rectal temperature from 98.4--I wonder why not 98.6--and divide by 1.5.

This is incredibly easy.  It is so easy that even Suzanna Doughcoup ought to be able to do it.  Upon her arrival, I had rehearsed the line that she could go to the Devil and that I would go back to bed.

There was a gentle knock on the door.  I opened it prepared to deliver what seemed to be a delicious line.

But it wasn’t her.  It was one of her loyal assistants who I knew would stay until I went with him.  I packed my line back up--along with my thermos of coffee--and got into the squad car with him.

It is only about five minutes from my house to the police station.  It’s not that I live particularly close to the police station; it’s just that nothing is more than about five minutes from the police station.

But, in those five minutes, I began to wonder about the simplicity of the Glaister Equation. It doesn’t take into account the environmental temperature and it doesn’t take into account the surface area of the body. Both of these are important factors in Newton’s Law of Cooling. 

As we got closer to the morgue, this worried me less and the rectal part worried me more.  I suppose that messing with a dead guy’s fanny is less awkward than messing with a live one’s as there would need to be fewer apologies afterward, but still: eeeewwww.

That last syllable was coursing through my brain when I set foot into the morgue and looked toward an autopsy table.  It was covered with the classic white sheet I expected, but there wasn’t nearly as much under that white sheet as I thought.

My first thought was “woman” because women are smaller.  But this was really small.  It was so small, in fact, and had such a shape that I shook my head in wonder at my earlier estimation of woman. 

I turned to the patrolman beside me.

“Armadillo?” I asked.

“Armadillo,” he confirmed.  He did so, amazingly, with a straight face. That straight face was put a lie to when he stepped out of the room and began to guffaw.

I stood there by the table asking why, why, why until Detective Doughcoup showed up.

“Glad that you could make it,” she said, “but I am sorry we don’t need you.”

“What?” I was to furious to go beyond monosyllables.

“Yep,” she said, “someone attempted to rob a liquor store and they ran over this armadillo when they made their get away. I thought that a time of death calculation would help establish the time of the crime, but then they told me the clerk had looked at his watch.”

I looked at my watch as a preparation for establishing her time of death, but decided I’d rather go back to bed instead.

“If you’d like to do it anyway, you are welcome to,” she said.

I looked at the thermometer and at the lump under the sheet.

“Take me home,” I begged.

“Have it your way,” she said.

The Power of Goodness

By Bobby Neal Winters

The other day I went out to have breakfast with a friend at out of our local downtown diners.  These are places wherein the 1950s still linger not only around the edges but dead center as well.  Every table is furnished with salt and pepper and a container of sugar.  Each sugar container has a cracker in it to absorb moisture and prevent clumping.  Some of those crackers look like they are left over from the 50s as well.

My friend, having reached the age where he has to monitor his food closely, ordered only oatmeal with cream and brown suguar.

“I still allow myself the cream and brown sugar,” he said with a bit of a smile.  “I enjoy what I can when I can.”

For my part, I ordered two eggs over medium with ham and wheat toast.

“Wheat toast,” he teased me. “That’s practically health food.”  

He’d directed his remark to the room at large and a patron who was eating alone at the next table enjoyed the remark in particular.  It turned out that he was my friend’s pastor and my friend made introductions.

After he’d introduced the other fellow as his pastor, he he introduced me by name and then by vocation.

“He teaches math and the university,” he said.

This can be awkward as some folks seem to feel obligated to disparage my life’s work on such occasions.  I have had other clergymen on such occasions tell me how much they hated math.  What if I would respond that I hated God?  Not that I do, but you get my  drift.

This fellow was different though.

“You don’t say?” he said.  “You know, even though math wasn’t my best subject, it amazes me.”

“Oh?” I asked. Having been jaded by years of libel, I was unprepared for this graciousness.

“Yes,” he said.  “Recently something that happened at my church brought out how surprising math--and people--can be.”

He’d said that he’d been serving the same church for over twenty years and knew every personality in it.  It was a church that was full of characters.  There were a couple of ladies of the church who were very much in the “church-lady” mold. This is a stereotype made famous by Dana Carvey on Saturday Night Live, but it didn’t arise from a vacuum.

“These are good, generous people,” he said. “I don’t want you to misunderstand me.  But their goodness carried an unpleasant overtone of pride.  They were proud that, even though they gave charity, they never needed it themselves.

“I tried working against this in my own subtle way by preaching about receiving gracefully, but, as with so many of my sermons, both subtle and blunt, it went right over their heads.”

Each of the women, it seems, prided themselves on giving more than they took.  Indeed, each claimed, loudly, in public, and often, that they gave twice as much as they received.  This all begin one day when the two women, Lois and Eunice, were together.  One of them, let’s say it was Eunice though it doesn’t matter, found a penny and, not wanting to seem greedy, gave it to Lois.  

“I don’t know what would’ve happened if they’d been alone,” the minister said, “but they weren’t.  Other people were watching, people who’d heard the loud proclamation of giving back twice as much as had been received.”

The next day, Lois gave two cents to Eunice.  The day following that, Eunice gave Lois four cents.

“It became a game with them,” the minister said. “They would laugh and joke. But the power of mathematics is like the power of God.  It works in small things.”

After seven days had passed, the amount of money going from one to the other was one dollar and twenty-eight cents.

“It was all in pennies,” said the minister. “They continued to scrape up pennies wherever they could find them.”

At first they had just passed the pennies back and forth in zip-lock bags, but on the ten day one of them switched to 2-liter pop-bottles.

“There were ten dollars and twenty-four cents worth of pennies and it filled the 2-liter bottle almost a quarter of the way full,” he said.

On the twelfth day, the bottle was almost full and on the thirteenth day they had to go to two bottles.

“This took on a nastier sort of tone on the fourteenth day,” he said. “That was the day they broke $100. My congregation is not rich, and $100 is at least a day’s pay to most of them. That day one gave the other $163.84 in four 2-liter pop bottles.  I saw the exchange take place. The bottles were from a generic brand of pop, and the look that the one gave the other wasn’t a look of charity.”

I was beginning to doubt the minister a little.  In fact, I had kind of doubted him from the beginning because this is an old mathematical chestnut.  It’s usually done with standing the pennies on the squares of a checkerboard.  You put one penny on the first square, two on the second, four on the third, so that when you get to the twentieth square the stack of pennies was a mile tall.

As he was talking, I’d surreptitiously taken out my smart phone and looked up the physical dimensions of the penny.  I’d worked out even the volume of that many pennies and his numbers of 2-liter pop bottles seemed plausible.  

But something had to give before too much longer.

“On the fifteenth day,” the minister said, “the pennies weighed as much as Lois’s husband.  On the sixteen as much as Eunice’s, and he was six-foot eight-inches tall with a love of ice cream.”

The minister sort of smiled at the last, but then grew more serious.

“They were now beyond their savings and had begun to pawn family heirlooms,” he said. “They no longer met face-to-face but just sent their husbands back and forth with the pennies.”

At this point, I gently broke in.

“So how does this all come out?” I asked.  “This couldn’t’ve gone on much longer, much less for ever.  What happened?”

He smiled at me in a sad way.

“You are right,” he said. “The mathematics controlled this.  It make me think of Calvinism in some way, but I digress.  It stopped after nineteen days.  On the nineteen day, Lois’s husband was exchanging $5242.88 with Eunice’s husband.  It was in 116 2-liter pop bottles and weighed almost 3000 pounds.  As I mentioned, Eunice’s husband had a love for ice cream and it proved to be his undoing and he dropped dead of a massive heart attack during the exchange.”

It was at that point, apparently, that Lois’s husband realized he’d never had that much cash as his disposal all at once in his whole life, so he called up his old high school girl friend Tamara, exchanged the pennies for twenties, and headed to Vegas.  

“He hasn’t been around since,” the minister said.  “Eunice and Lois have been comforting each other over their respective losses.  I hope they have learned something about the sin of pride.”

“Or the geometric progression,” I said.

“At least,” the minister said.

The Arrow of Fairness

By Bobby Neal Winters

I have passed the age of eating chili late in the day without having to suffer dire consequences.  I am also beyond the point where I can watch very much in the way of politics right before I go to bed.  To be so stupid as to combine Frito Chili Pies with onions while watching election returns is almost beyond imagining, but I don’t have to imagine it because I actually did it.

The election coverage kept me up late, and, when I did go to bed, it took me a while to go to sleep, and, when I did go to sleep, it was one of those wakeful sleeps where one is not sure whether one is awake drifting into sleep or asleep rising toward wakefulness.  It was in such a state that I began to dream.

In my dream, I was in a modest, rural community.  The roads were dirt and the buildings were unpainted wood, stained gray by weather.  In the center of town there was an official looking building, still of unpainted, weathered wood but official none the less.

It was clear there was an election going on, because there were signs everywhere touting the three candidates that were running: Abraham Lincoln, Bobby Kennedy, and Calvin Coolidge.  I recognized them by their pictures because the names on the signs were simply Abe, Bob, and Cal.

I walked toward the official looking building and entered in to it.  There inside was a glowing, golden cube that sported a 21-inch touchscreen monitor upon which was written: “Election results here.”

I was asking myself what this could be when, in the manner of dreams like this, there was a voice at my elbow with an answer.

“That’s our guaranteed fair voting machine,” the voice said.

I turned to see a man dressed in old style.  He was wearing a black “boss of the plains” hat. This is a round hat with a broad, round brim that curves down.

His shirt was white but everything else was black: coat, tie, and pants.  His tie was actually a neck cloth and tied into a horse collar knot.

The word ‘fair’ caught my attention.  It means so many things to so many people.  When mathematicians refer to a fair coin, they mean one that will come up heads or tails equally often, but, when most people use it, they mean something different.

“A fair voting machine,” I echoed. “Does that mean it counts every ballot correctly?”

“Oh,” he said, “it does much more than that. Come and see.”

I followed him and he showed me a ballot with three names on it: Abe, Bob, and Cal.  The voters were instructed to mark each of these with a 1, 2, or 3, according to preference with 1 being most preferred and 3 being least preferred.

“By fairness, we mean the following two rules are followed,” he said. “The first rule is that if everyone prefers one candidate to another, say Abe to Bob, then Abe will be rated above Bob in the election.  The second is that this will remain unchanged even if the voters change the relative ratings of other candidates.  That is to say, moving Cal higher or lower will not effect whether Abe is preferred to Bob or not.”

I nodded my head as that seemed fair to me.

My stomach rumbled and called me briefly toward--but not actual to--wakefulness and I remembered something vague about this kind of situation, but I rolled over and pushed myself into a deeper sleep.

The town was there as before and the man, who I began to call ‘The Parson,’ was standing there with me.  In front of us, queueing up to the voting machine, were the people of the town.  All of the men were dress in the same manner as the Parson and all of the women where dressed in gray woolen dresses.

“What are they lined up for?” I asked.

“They are lined up to vote,” the Parson said.  “They are lined up in the order in which they have voted from time immemorial and always shall vote.  They put there ballot in and walk past and allow the next to vote.  Then the machine gives us the town’s preferences in order.”

“So you get more than just a winner?” I asked.  “You get the town’s collective preference?”

“What you say is so,” the Parson replied.

“I am curious,” I said.  “Would you mind if I did some experiments with your machine?”

The Parson’s expression then changed from the grim one he’d been wearing into a smile.

“It is your dream,” he said. “Go for it.”

I turned to the line of voters and began to speak.

“The first experiment will have several parts,” I said.  “The first thing I want you to do is to mark your ballots with Bob as your third, that is last, choice and then vote.  If your machine is working as you say, this should give us that Bob loses.”

The line, dourly dressed men and women alike, filed past the machine inserting their ballots.  When all were done, I looked at the results on the machine’s screen: Bob was last.

I’d never doubted the machine was accurate, I just wanted them to have an initial state in which a particular candidate, Bob in this case, was last.

“Okay,” I said, “the next part of the experiment will be more complicated.  We know that if everyone lists Bob as their first preference that he will be the winner.  What I want to do, is to ferret out a particular key voter.  To do this we will have to vote many times.  The first time, I want the first person in line to switch Bob from last to first while preserving the order of the other two candidates and for everyone else to leave their ballots alone.  We will then check to see whether Bob has been moved from the bottom.  

“If Bob hasn’t been moved from the bottom,  the first and second people in line will then both move Bob to the top while leaving Abe and Cal with the same relative position to each other as before on their ballots while all the rest of the voters’ ballots will remain as in the initial phase of the experiment. After everyone has voted, we will check again to see whether Bob has been moved from the bottom.

“We will continue in this manner until, as must happen, Bob has been moved from the bottom. Do you understand?”

Much to my surprise, they all nodded that they did understand.  

The process began and the whole line of voters modified their ballots and voted and revoted numerous times, each time checking whether Bob was still the loser.  At some point, the Parson checked the winner and called for a halt.

“It’s done,” he said.  “Bob is no longer the loser.  It was our own brother Nimrod who did the deed.”

“Nimrod,” the rest of the voters chanted in unison.  It was kind of spooky.

I nodded my head.  

“I will now tell you something,” I said.  “Not only is Bob not the loser; he is now the winner.”

It was the Parson’s turn to be spooked.

“How do you know that?” he asked.

“Just mathematical reasoning,” I said.  “For the sake of argument, assume that Bob is not the winner but, say Abe, is.  Then Abe beats Bob and Bob beats Cal.  Now pretend that Nimrod and everyone before him vote for Bob, Cal, and Abe in that order and everyone after them in line to vote Cal, Abe, and Bob in that order.  The relative positions of Bob and Abe haven’t changed so Abe beats Bob.  Similarly, the relative positions of Bob and Cal are still the same so Bob still beats Cal; therefore Abe beats Cal.  The kicker is that every voter prefers Cal to Abe, so Cal must beat Abe.  As we can’t have Abe beating Cal and Cal beating Abe, the only alternative is that Bob must be the winner.”

Fear was in the Parson’s eye.

“Truly, you are a sorcerer,” he said.

“Sorcerer,” the votes chanted as one.

At that point my wife started shaking me.

“You are having a bad dream,” I heard through the haze.

I couldn’t rouse myself but turned over to my other side.

In my dream, the Parson was there again.

“You said you had experiments, Sorcerer,” he said.  “That means there is more than one.  Tell us your second, as the night grows short.”

“Okay,” I said.  “I will now show you that Nimrod can always dictate Abe and Cal’s relative positions with respect to each other.”

“Show us, Sorcerer,” the Parson said.

I noticed that Nimrod was now wearing a little square mustache on his upper lip. Everyone else was clean shaven.

“We will vote only four times,” I said.  “This time Nimrod will vote so that he prefers Abe to Cal.  He can vote Bob in any position he wishes and everyone else can vote as they please.

“In vote 2, change the ballots in vote 1 so that everyone who votes before Nimrod has Bob and the top of the ballot but that Nimrod and everyone after him has Bob at the bottom.  The relative positions of Abe and Cal to each other should remain the same on each ballot.

“In vote 3, change from vote 1 so that Nimrod and all of the voters before him put Bob at the top and all those after Nimrod put Bob at the bottom.  In other words, this is just like vote two, but Nimrod has moved Bob from last to first.

“Finally, in vote 4, everyone before Nimrod votes exactly as they have in two and three, Nimrod votes Abe over Bob over Cal, and everyone after Nimrod votes as they did in two and three. Now don’t tell me what happens in vote 1.”

They did as they were told, and the following happened.  Bob lost in vote 2.  This had to happen because Nimrod didn’t vote for him  We knew from the first experiment that Nimrod had to be the voter to raise Bob from the bottom because the “fairness” of the magic cube guarantees changing the relative positions of Abe and Cal on any of the other ballots will not change the position of Bob in the final vote.  We know that Bob wins in vote 3 by the same reasoning.

In vote 4, we know that Bob beats Cal because of experiment 1 and the fact that we get to ignore Abe because “fairness” assures us that only relative preferences matter.  We also know that Abe beats Bob by the same reasoning.  So Abe beats Bob who beats Cal, and it follows that Abe beats Cal in vote 4.  

Since only the position of Bob changed from vote 1 to vote 4, I know that Abe beat Cal.  I told them.

“Yes, Sorcerer,” the Parson said, “you are correct.”  He glanced toward Nimrod, who seemed to be growing in stature with the rest.  

I began to feel sweaty.  I heard my wife say, “You will never learn.”

I turned over yet again.

This time I was in the clouds.  I began to think.  Is Nimrod so special?  I could have done the first experiment with Cal being on the bottom instead of Bob and then Michael might have been able to dictate the relative positions of Abe and Bob.  I might’ve done it with Abe being on the bottom and turned out with Paul being able to dictate the relative positions of Bob and Cal.

But then Micheal could vote for Abe over Bob and Paul for Bob over Cal, so Abe would beat Bob who would beat Cal, but then Nimrod could come along and vote for Cal over Abe.  It wouldn’t make sense.  I turned it over several other ways in my head and the only way it made any sense was for Nimrod to be the sole dictator.

Suddenly, the clouds were gone and Nimrod was there with the Parson and the other voters with pitchforks and sickles.  

“The Sorcerer is there!” the Parson screamed.  

“Get him,” Nimrod said.

And they were coming for me, and then I heard bells ringing.  It was the alarm clock.

In the shower, I recalled consciously what I’d been dealing with in my subconscious all night: Arrow’s Theorem, which is named after the economist Kenneth Arrow.  Given any sort of voting scheme with more that two candidates that attempts to order them while keeping the two the fairness criteria described, there must be one voter who will be able to dictate the outcome, which, paradoxically, doesn’t seem fair.

It’s enough to keep anyone awake at night.

[Based on a proof by John Geanakoplos]