Redneck Math

The Shoulders of Giants

2011-08-22T15:37:00.001-07:00

The Shoulders of Giants

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.--1 Kings 7:23

The verse above frequently comes up in arguments between atheists and believers. It implies that the value of the mathematical constant Pi is 3 as opposed to 3.14159... .

Some atheists use it as an example of a mistake the Bible which proves that the entire Bible is a lie and that, therefore, Christianity is a sham. Believers, when they deal with this, have been known to erect huge houses of cards in order to preserve the truth of the Bible on the atomic level.

Let us put the polemics aside for the moment and just take a look as this issue.

Isaac Newton, the physicist and mathematician, is credited with saying that if he saw further it was because he’d stood on the shoulders of giants. Newton had discovered the Law of Gravitation that bears his name along with the part of mathematics known as the Calculus and, in doing that, had benefited from the work of ancient Greek minds such as Archimedes along with the coordinate geometry of Rene Descartes, who had himself benefited from algebra that had come to the West from the Arabs.

While the typical high school or college student might wish that these people, be they giants or not, had never existed, without them we would be without many of the modern amenities that we take for granted, particularly those amenities that require having satellites in orbit. The mathematics is complicated but very useful. It is also very beautiful.

Newton referred back to the geometry of the Greeks. It is somewhat ironic that, although Newton did invent the Calculus and the Calculus is the way we prove Newton’s results today, he demonstrated them himself with synthetic geometry in the manner of the Greeks. The Greeks mathematics for its own sake; they were interested in it for its beauty. They pursued geometric truths because they found them to be beautiful.

Again, this is ironic because, when you are in the business of building a civilization, geometry is very useful. If you are going to have a civilization, you will need at least one city to prove you are civilized. That city will need palaces and temple and streets. Geometry is necessary for the construction of all those things. If your civilization isn’t going to starve, you will need a calendar in order to keep track of time. For a good calendar, you are going to need astronomy, and in order to do astronomy well, you are going to need geometry.

When the Greeks were doing geometry, they ran into the number Pi. The number Pi, as you may recall, is defined to be the circumference of a circle divided by its diameter. The word circumference comes from the Latin and obscures the Greeks referred to the circumference as the perimeter. The word perimeter begins with the letter pi in Greek from there is it easy to see why we call it Pi.

To me, it is an amazing fact that this ratio of circumference to diameter is the same regardless of whether the circle is ten inches across or ten miles across. While my students have doubted my word on many things--in particular on the value of doing daily homework assignments--they’ve never doubted me on this thoroughly amazing fact. The ratio of the circumference to the diameter of the circle is constant.

It is even more amazing that they believe this because it’s not true some other models of geometry. Pi isn’t constant in either hyperbolic or spherical geometry. You may have heard the old joke that if you tell people there are a billion-billion-billion stars in the universe they will believe you, but if you tell them the paint is wet they’ll have to touch it.

Pi is interesting mathematically because it’s an irrational number. Irrational numbers are typically explained in school to be numbers whose decimal representation never repeats. And that is true but it makes use of a way of expression numbers that was foreign to the Ancients. The Ancients didn’t have decimals. They didn’t really have fractions the way we have them. They worked with ratios.

A rational number is a number that can be expressed as a ratio of two whole numbers. Pi cannot be expressed that way so it is irrational. The Pythagoreans--those clever Greeks again--proved a result that in modern language and concepts boils down to the fact that the square root of two is irrational. This is a comparatively easy exercise when one has modern notation. This irrationality of Pi does not have such an easy proof.

I’ve worked through a particular proof that Pi is irrational and satisfied myself that I understood every step and had a clear enough ideal of the “big picture” to present it to a class of seniors. When I did this, they had the same expression on their faces as I presented it as my cat has when he watches me program the DVR. It requires quite a bit of mathematical machinery and is not something the Greeks would’ve come up with. The Greeks didn’t come up with anything else either.

The Ancients, Archimedes among them, did come up with ratios that approximated Pi. None of these was exact, of course, but only an approximation. As I have said, there is no exact such ratio.

Let us now talk about the Biblical context in which this arises. Solomon had hired builders from Tyre to built a temple to YHWH and while they were at it to build a house for him. This is an interesting use of the word house as palace would be a better description. This makes me think the PR people were already at work in those days. This section of the Book of First Kings is describing the palace in what I as a reader consider to be excruciating detail. In this, the passage has much in common with modern writing of the same type.

In any case, this molten sea is a round pool. It is described as having a circumference of 30 cubits and a diameter of 10 cubits. This would give us Pi=30/10=3, which we mentioned earlier.

From the modern point of view, this is redundant. We know that when you give the diameter, you also give the circumference and vice versa. The folks writing this description either didn’t know this or were wedded to redundancy. Given the style of many of the drier patches of scripture, there is a case to be made for the latter, but I believe they just didn’t know. Solomon had to hire this construction done. Likely, they simply didn’t have the construction expertise available in Israel. Without a well-developed practice of construction, they simply didn’t have the geometrical chops to know.

Now I would like you to do a thought experiment. Act as if you are going to measure this yourself. I would go to the hardware store and get a tape measure to measure the circumference and then measure the diameter. I would have to use the tape measure because the pool is round. Tape measures hadn’t been invented in those days and neither had hardware stores. Likely they would have used a piece of cord and then measured the cord. They would’ve measured it in cubits.

A cubit is an ancient form of measure that is based on the length of a man’s forearm. I belief I am safe in saying that forearm length was as variable in antiquity as it is now, so in order for this method to be useful, they would’ve probably had to set a standard cubit at least for a given job. That having been said, we can estimate the cubit to be about 18 inches. This puts the pool at being about 180 inches in diameter. Using the exact value for Pi, this would make the pool about 31 and a half cubits in circumference.

In other words, the description given in scripture is about one and a half cubits off. In my mind, this was an amount that they would’ve noticed in the measuring. They could’ve put down the correct number quite easily, I have to assume they didn’t because they just didn’t care.

They didn’t care because giving the correct value of Pi was not the purpose of this text. I challenge you to read the seventh chapter of First Kings. I double-dog dare you.

This text is about the opulence of Solomon’s Palace. LIke the Russia in the cable TV commercial, Solomon could’ve said, “Opulence, I has it.” Those of you who are Bible scholars know that Solomon taxed Israel to the point of ruin and his son finished the job. In this text, you see part of where the money went.

Pi, schmi, Solomon had a swimming pool that was 15 feet across 47 feet around and seven and a half feet deep. In a desert, if you got this you can say, “It’s good to be King!”

In conclusion, I’ve got to say that for me this is a non-issue, but I do understand there are those for whom it is. In my opinion, the Bible is not about Pi. It is about the fact that we all stand on the shoulders of giants and that we should never forget that.

The Laziness Index

2011-07-28T11:59:00.000-07:00

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

2011-07-24T20:01:00.000-07:00

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_Ris 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

2011-07-19T13:12:00.000-07:00

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

2011-07-07T12:51:00.000-07:00

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]

A morning at the Farmer’s Market

2011-07-04T12:34:00.000-07:00

A morning at the Farmer’s Market

By Bobby Neal Winters

While I am not a fitness nut by any means, I do habitually take a daily walk. I am also an early riser even on Saturday mornings. Sometimes, when all of the stars are aligned correctly, I can make these two things work together and get over to my town’s Farmer’s Market which is only a stroll of ten blocks from my house.

That is the way it was on one particular Saturday morning that I am going to tell you about. I got up, dressed, and went immediately outside. It was not the cool weather one gets on spring mornings, but neither was it as hot as it was destined to get later in the day.

The first part of my walk was in the shade of the trees, but out Farmer’s Market is in the middle of town where trees are indeed scare. I missed the trees as it appeared to be getting warm rapidly. Indeed, as I came to Broadway, which is the principal north-south thoroughfare in my town, I saw heat rising from it.

Looking across Broadway, the Farmer’s Market looked busy and the heat waves rising from the street obscured it. As I crossed the street, there was a gust of wind that carried sand along with it. The sun seemed to jump in the sky and it became noticeably warmer. As I finished crossing the street and entered the Farmer’s Market, it seemed different.

I didn’t see the usual mix of yuppies and would-be-hippies that typically populate the Market. The vendors seemed to be of a more middle-easterly mix than usual. I thought that perhaps today they had a special international theme and that maybe some kids at the university were involved--or there parents as some of the vendors seemed a little too old for school.

The impression that it was a special day was reinforced by the palm trees that surrounded the square. And then there were the tents which seemed to be made from natural fibers rather than synthetic and were a good deal dirtier than usual.

The offerings at the various booths were different as well. There were the booths that offered the usual fresh vegetables, of course, but whereas there is usually a booth that sells breakfast burritos (yum), there was instead a booth that sold goat meat roasted on a skewer. You could see the whole skinned goats hanging by their heels with varying amounts of flies.

And instead of the booth of the fellow that sells homemade bread while touting its health benefits, there was a fellow with a brick oven selling freshly baked bread in flat, round loaves. I wondered how they got that brick oven in during the week. Someone had spent a lot of money on this.

Then there were the smells. In addition to the roasting meat and the baking bread, I caught the smell of spice. Cumin? Curry? I couldn’t tell, but it was over-powering to me. It hooked my nostrils and dragged me the length of the Farmer’s Market to a booth that was surrounded by bags of spice that were hip-high and two feet across.

Within the booth was an olive-skinned man with wrinkles as deep as the Grand Canyon in his weathered face. He was busy about the business of selling spice, and he was using a simple, but ingenious, device that deserves description.

At first glance, it looked like Justice’s Scales. There was a horizontal stick, hanging from a fulcrum with conical containers on either side of the fulcrum. When I say conical, you might think snow cone, but I want you to rethink that slightly. The were pointy end down because they were being used to hold measured amounts of spice, but they much wider and flatter like the hats worn by farmers in southeast Asia.

I saw there was more to it than just a balance when I saw a customer buying spice. The customer pointed at a particular bag and said something to the merchant. I could not hear him above the din, but he clearly held up five fingers. The merchant then took a nondescript cup, plunged it into the back of spice, and retrieved it. Then he carefully--and in full view of his customer--scraped off the top of the cup until the spice was level. He then carefully poured it into one of the conical pans on the balance. It was then I noticed that the stick between the two pans was filled with equally-spaced notches.

The pan in which he put the cup of spice was one notch away from the fulcrum. He then did something that surprised me. He moved the other pan to a notch that was four notches away from the fulcrum on the other side. I thought that I knew what was happening and expected that he would move it five notches. I watched expectantly.

After getting the pan in the right spot, he measured in cups until it balanced. There were exactly four times the amount of spice in pan on one side than in the other. He then poured all of the spice from each pan into a bag that was proffered by the customer. In return, he accepted some coins offered by the customer. The coins looked pretty cool. I wondered if they were tokens that were being sold in another booth somewhere.

Putting his second pan four notches from the fulcrum made sense to me now. It was the most efficient way to perform the transaction. Four plus one is five after all.

I was about to leave to find the booth that sold those tokens, when another customer came. He also said something that couldn’t be heard above the din as he flashed both hands twice and then followed that with a peace-sign.

I took this to mean that he wanted twenty-two measures of the spice. This was confirmed when I saw the vendor reach down into a bucket that I’d not noticed up until that point and draw out some tokens and lay them out carefully in two lines of eleven. The tokens were disks like coins, but they had slots cut mysteriously into one side.

As I was counting to eleven, he was scooping out exactly one measure of spice as before and putting it into one of the pans. He then moved the pan on the other side of the fulcrum to another notch and surprised me again. I was prepared for him to move it to the twenty-first notch from the fulcrum, but then I noticed there were only seven notches on that side. He didn’t move it to the seventh notch either. He put it in the second notch.

I was confused, but I was also intrigued. I continued to watch.

He then put spice in the two notch side until it balanced. That side now had double the spice as the first. Then he then turned his attention back to the two lines of slotted tokens. He deftly swept one of the lines back into his token bucket and then laid out his tokens into two lines again. This time, as there were eleven tokens left, there was one line of five and one line of six. He removed the extra token from the line of six and fitted its slot onto the double pan.

He then slid it back to being one slot from the fulcrum. Then he put spice in the other side until it balanced. Then he took the pan that had the token on it and dumped the spice in it into his customer’s bag and put the token back into the token bucket.

He then placed the now empty bucket back into the doubling notch, as I was now beginning to call it. It looked a bit more well worn than the others. He fill it until it balanced against the other side. I was keeping track. The other side had two measures in it, so that meant the doubled side had four. He took the five tokens that he had left and lay them out in two rows; one with two tokens and one with three. He took the extra one from the line with three and fitted its notch onto the doubled side.

As before he moved it from the doubling notch to single notch and added spice to the other side until it too had four measures. He took the pan with the token and dumped its spice into the bag and put the token back into the token bucket.

Again, he put the empty bucket back into the doubling notch, and filled it with spice until it contained eight measures. He then swept one of the lines of two tokens back into the bucket and lay out the remaining two side by side. There was no extra token, so he didn’t fit it on the edge of the pan. Neither did he dump the spice from the double pan into the customer’s spice sack. He did, however, put the pan containing eight measures to be a single notch from the fulcrum and balance the pan on the other side to contain eight measures.

He took his two remaining slotted tokens and fitted one to each of the pans. He then dumped each in turn into his customer’s waiting spice sack, adding a total of sixteen measures of spice to the sack.

As I said, I’d been keeping track of the spice dumped into the sack. He’d dumped in two measures and then four and then sixteen, making a total of twenty-measures, just as the customer had requested.

I was fascinated--and I just had to get some of that spice because the smell was driving me crazy. I decided that I would get some of those coins that everyone seemed to be using to buy the wares, so I started looking around for a booth that sold them. I tried asking the people, but they didn’t understand me.

One thing I need to mention is that the Farmer’s Market abuts the south side of a bank parking lot so I figured they must be selling the coins at the bank. In order to pursue this theory, I headed north.

It was then there was another gust of wind and sand, and the sun seemed to jump in the sky again. I went to the bank, but it closed, and when I went to the Farmer’s Market again, it seemed like it always had before. I couldn’t find any of the thematic booths.

I walked back home and took a nap.

Waking back up, it had all become clear to me. The man in the spice booth had been using a method called the Russian Peasant Algorithm or Egyptian multiplication. While it can be used as multiplication, it is really a method for representing a number as a sum of powers of two. You can try it yourself by using two different kinds of coins.

In the table below, I duplicate the process by using #-signs for my number, which was represented by slotted tokens in the Market, and &-signs for the measures of spice. I mark the times that dividing by two left a remainder by putting a $-sign in the far right column.

##### ##### # ##### ##### #	&
##### ##### #	& &	$
## ###	&& &&	$
# #	&&&& &&&&
#	&&&& &&&& &&&& &&&&	$

Notice that if you a count all of the &-signs whose column is marked with a $-sign you will get twenty-two, the original number. Now let me modify this table to illustrate another principal. I will replace the original & by the weight of an amount of spice, say 213 grams.

##### ##### # ##### ##### #	213
##### ##### #	426	$
## ###	852	$
# #	1704
#	3408	$

Notice, each of the numbers is just the double of the one above it. If I add up the numbers in the rows marked with $-signs, the sum is 4686. Note that this is equal to 22 time 213.

While it is a perfectly legitimate way to do multiplication, it seems magical because it eschews the notation that we use in the decimal system. The Russian Peasants who used a similar method didn’t know the decimal system because they were illiterate. The Egyptians didn’t because it hadn’t been invented yet.

Soccer Balls and Sequins

2011-07-03T14:12:00.000-07:00

Soccer balls and sequins

By Bobby Neal Winters

I was sitting by the soccer field stupefied.

This is not an unusual condition for me, that is to say, neither stupefaction nor watching a soccer game. My daughter plays soccer so I am frequently at the field watching her. And there are also frequent intervals on any particular occasion when I am astonished or shocked. But I do find myself sitting by a soccer field in such a state more frequently than random chance would merit. This is because I have no understanding of the game.

But on this occasion, my stupefaction had nothing, or at least very little, to do with the game at hand. I say very little because my confusion with aspects of the game had led me to withdraw into my own mind while staring at an object in the immediate environment. As I was by a soccer field, the object at which I was staring turned out to be a soccer ball. I was mesmerized by it. I was noting how its surface was covered with hexagons and pentagons; twelve pentagons and twenty hexagons. I am not sure where my musings would’ve taken me if my reverie hadn’t been interrupted by an interloper.

The interloper was about two and a half feet tall. Determining his age is problematic. I figured he had to be at least two years old to gather that much dried mucous in his nostrils. I decided he was somewhere between too old for that loaded diaper he was wearing and too young for that sharp pair of scissors he was carrying.

It was the sight of the scissors in the young child’s hand that really woke me up. While I usually keep my distance from the children of strangers, I make an exception to this when I believe the child is unattended and I believe the child is in danger. I was about to intervene when the child’s mother arrived on the scene.

“There you are, you little scamp,” she said. “Did you find yourself a soccer ball?”

He is say anything, but he rubbed the soccer ball and giggled.

“Hehehehe,” he laughed. It came out with such rapidity it sounded like a little laughing machine gun.

“So are you going to play with it?” she asked.

“Hehehehe,” he replied.

I was on the verge of thinking to myself how cute this was when the scissors plunged into the soccer ball. My eyes glanced up toward his mother to see her reaction.

“Oh my goodness,” she said. “What a strong little boy!”

I looked side to side to note the reactions of the folks around me, and they were all looking at the game, of all things.

“What do we do now?” she asked him in a baby-talk tone. His reaction was immediate as he began cutting the ball into pieces.

By this time, she’d noticed that I was watching, and I’d noticed that she’d noticed that I was watching, so, in an attempt to keep things from getting awkward, I thought I might start a conversation.

“So your son,” and I was sure of the son part because the loaded diaper was hanging a bit low, “your son likes to cut up soccer balls?”

“Oh, yeah,” she said. “It keeps him busy while his big brother is on the soccer field playing.”

I glanced out an the soccer field, but I didn’t see any of the twelve-year-olds running around in diapers, though there was one with a tattoo.

“I wouldn’t want him to get run over,” she continued. “Plus, it helps me with my hobby.”

“Hobby?” I echoed stupidly.

“Yes,” she said. “I glue them back together and turn it into a craft. I’ve got one hanging from my review mirror. See?”

As she spoke she pointed toward the street where I saw an ancient Volkswagen van with what was apparently won of her re-constituted soccer balls hanging from the rear-view.

About that time, I heard laughter from the boy again.

“Hehehehe,” he giggled. “Done.”

I looked in his direction again, and there in front of him were a pile of pieces of what had been a functioning soccer ball. He had--and in short order I must say--cut the ball into a small number of irregular polygons.

“So how do you do your craft?” I asked.

She seemed pleased by my inquiry.

“Oh,” she said. “It’s real easy. I use a hot glue gun and glue the pieces back together along one of the cuts he’s made. Then at places where several of those cuts come together I put a big shiny sequin to cover the hole.”

She paused for an instant as if getting an idea. Then she smiled.

“I got an idea,” she said. “I’ll just run over to the van and get one to show to you. I’ve got lots of them.”

She left her son, who was now digging in the dirt with the scissors, went to the van, and fetched back the reconstructed soccer ball.

It was just as she said. The pieces of the ball were joined together by hot-glue (now cold of course) and there was a big sequin where the ends of the seems came together. Absently, I started counting: nine sequins, fifteen seams, and eight pieces. I must have been doing this out loud because the woman commented on it.

“What did you say, mister?” she asked.

“Oh, I was just counting the sequins, seams, and pieces,” I said. “I do this because of Euler’s formula.”

“Oiler?” she said.

“That’s right, Leonard Euler, a Swiss mathematician discovered that it you slice up a sphere in the way your son did that the number of the vertices minus the number of edges plus the number of faces is always equal to two. That is to say, sequins minus seams plus pieces is two. This works as long as the pieces are topological disks.”

“Topo-whatical whats?”

“Topological disks,” I said. “Though it took many years to create the mathematical machinery to put it into those terms. The way we say it now is that the Euler characteristic of the sphere is equal to two.”

By this time, the woman and her son were both looking directly at me, looking somewhat stupefied themselves. If I let that look bother me, I would have to give up my career teaching math. I continued undaunted.

“Euler used this formula to prove there are only five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.”

“Eye-eye-eye-caws a head run,” the boy said.

“That’s right,” I said. “And it’s funny you should mention that one in particular, because if you snub an icosahedron, that is do some careful clipping near its vertices, you obtain a soccer ball, which is a very pretty example of what is known as an Archimedean solid, even without the hot glue and sequins.”

By this time, she had picked up her son, taken the scissors from him, and was nonchalantly moving away from me.

And that had sort of been part of my plan.

When the game ended, I took my family home and took out the pad I keep by my recliner for moments of mathematical thought.

In a Platonic solid, we assume that every face has the same number of edges; call this number m. We also assume that the same number of edges comes into every vertex; call this number n. In addition, every edge has two vertices and every edge is on two faces. If we let V be the number of vertices, E the number of edges, and F the number of faces, we have the following equations:

V-E+F=2

m F=2E=nV

(2/n)E-E+(2/m)E=2

This can be rearranged to 1/n+1/m=1/2+1/E, and, since E is positive, this yields the inequality 1/n+1/m>1/2. Now n and m must be at least 3, so we can by work out by elimination that (n,m) can only be equal to (3,3), (3,4), (3, 5), (4,3), and (5,3).

From this information, it can be shown there are a total of five Platonic solids: the tetrahedron having four faces, the cube having six, the octahedron having eight, the dodecahedron having twelve, and the icosahedron having twenty. Each of these obeys the formula vertices minus edges plus faces equal two.

The soccer ball, with its pattern of twelve pentagons and twenty hexagons, a total of thirty-two faces, has 60 vertices and 90 edges, and also follows Euler’s formula.

Too much pair of dice

2011-06-30T12:56:00.000-07:00

Too much pair of dice

By Bobby Neal Winters

Custer walked across the parking lot toward the tall building. The sun was setting and the lights on the side of the building were beginning to be more visible in the darkness. The colored lights, red, blue, and green, alternately flashed and shot up the side of the building.

Unknown to Custer, one hovering above his right shoulder and one above his left, were entities.

Those of a certain bent might call them spirits, others might refer to one as an angel and the other as a devil, and those of a more secular stripe might refer to them as manifestations of Custer’s bicameral mind. We will call them Poindexter and Jezebel and refer to them as entities, allowing the reader to interpret things as the reader might wish.

The closer the trio approached the tall building, the more Poindexter became concerned.

“That sure is an expensive building,” he said. “I wonder how they can afford that.”

Jezebel, for her part, was delighted. She had a natural aura about her and it pulsed in harmony with the lights of the building.

“Just like you to think about money,” she said dismissively. “They’ve designed everything so that it will be more fun!” With that she shot off into the distance, circling around a couple of pretty girls who were much too immodestly attired in Poindexter’s opinion. She then shot up, circled the neon sign on the front of the building that read “Casino” twice, and shot back down to her position above Custer’s shoulder.

Though Poindexter as an entity surely had no need of glasses, those who had seen flashes of him often remembered him as having horned-rimmed glasses. Has they had such a flash now, they would’ve seen him with those glasses slid down toward the end of his nose.

“You know,” he said with as much seriousness as he could muster, and he was capable of mustering seriousness by the truckload, “if you lose the first and last letters of casino what you have left spells ‘a sin.’ Maybe you ought to think about that.”

Jezebel, who’d only also been seen in flashes, would’ve subconsciously reminded Custer of a twenty-year-old version of his Grandma Vidalia. This might have bothered Custer a great deal had he been paying attention during his college psychology class, but he wasn’t because he was distract by a real twenty-year-old girl who subconsciously reminded him of his Grandma Vidalia.
In any case, Jezebel simultaneously shook her head and rolled her eyes upward.

“Whatever!” she said, making that last syllable last about ten seconds.

Unaware in any real sense of either of the entities around him, Custer flowed along with the growing stream of humanity around him until they left the growing darkness of the outside world and enter into the interior of the casino with its artificial stars dazzling him with their brilliance.

Once inside, Jezebel flitted around with delight. She buzzed past all of the slot machines with their whirling numbers, flew under dice as they themselves rolled along the tables, and orbited around the roulette wheels as they turned.

Poindexter, in stark constrast, grew more and more stern. As Jezebel hovered above a waitress who was exposing altogether too much cleavage both above and below and drew more of Custer’s attention, Poindexter prepared for one last stand: he would use pure mathematical reason.

Poindexter reached into Custer’s mind and stimulated a memory of one of Custer’s math class. This wasn’t too difficult because Custer’s math professor looked much like the way people remembered Poindexter looking.

Hovering above Custer’s right shoulder, Poindexter said the words, but in Custer’s head he remembered his professor saying them.

“The first thing you must know,” the words came out dead and dull, “is that a casino is in the business to make money, not to be fair. Whatever the game, the casino will always have an edge.”
Custer was dazzled by the spectacle about him, but never the less he echoed the words.

“Have and edge.”

“The games are arranged so that the house will win, on the average, just a little more than half of the time. Let us denote this edge by the Greek letter epsilon.”

Jezebel’s attention had settled upon Poindexter’s efforts.

“Epsilon,” she laughed. “You think that you are going to convince him with an epsilon.”
Poindexter looked daggers at her but continued.

“Epsilon must be chosen carefully,” he said. “If the edge is too much, then the gambler’s will not get enough positive feedback from winning, but if it is set too small the casino will not make enough money to pay it’s expenses.”

“Expenses,” Custer mouthed subconsciously.

“The casino,” Poindexter continued his lecture, “is able to take advantage of the Central Limit Theorem. With a large enough number of gamblers, the casino can assure itself a profit even if it has allowed itself only a small edge.

“For the sake of argument, assume that a casino has 1000 patrons during the course of a day and that each of these patrons gambles $20, stopping when they have gambled with it all, not necessarily lost it, just gambled with it.”

“Just gambled with it,” muttered Custer. His eyes were fixed upon another patron of the casino who was wearing cowboy boots and hat and cutt-off jeans he imagined as just one thread away from being a skirt.

Poindexter husbanded his strength and Jezebel circled the young lady’s halter-top.

“Also assume each gambler gambles at least 30 times and that the edge, epsilon, is equal to 0.05, that is to say, five percent.”

“Epsilon,” Jezebel rolled her eyes again.

“At the end of the day,” Poindexter/professor was triumphant, “the casino will have won $2000 dollars give or take $15. At the same time the average gambler will have lost $2, which isn’t much, but, what is more, 290 of those 1000 gamblers will come out ahead. Some of them might come out way ahead, and you can be sure that they will be put in the spotlight.”

Poindexter could see that Custer was thinking on his own now and listened to Custer’s subconscious.

“Only lost $2. In the spotlight.”

Custer smiled. This was bad, Poindexter thought.

“But don’t you see?” Poindexter was panicking. “If you gamble long enough, you will inevitably lose yourself. And you in particular won’t be able to gamble with discipline. You will just gamble your money until it is all gone, and even if you do win, you will just gamble until that’s gone!”

“Gamble ‘til it’s gone!” This time Custer shouted.

This caught the attention of a waitress in a very tight t-shirt who gave him a complimentary drink. Custer then walked over to a roulette table, plopped a twenty down.

“Red!” he shouted.

Jezebel, who was all aglow, flittered over to Poindexter’s side.

“Buy you a drink?” she asked.

“Make it a double,” he said.

A square deal

2011-06-28T19:44:00.000-07:00

A square deal

By Bobby Neal Winters

Jim looked at his daughter, Kristin, who was down on the floor working on her homework.

“What ya doing there, Sweetie?” he asked.

The girl was lying on her tummy with the book spread before her and the TV blaring in front of her. She craned her neck back towards her father and smiled.

“Geometry,” she said. “Wanna help?”

At the word geometry, Jim made a face.

“I can’t say I can help you much with geometry,” he said. “I could always figure it out and always thought I understood it, but I never seemed to get it to suit the teacher. ”

He’d always enjoyed doing his math homework, but whenever he turned it in, it always came back painted red. He’d leave out the T in his ledger proof or something like that. After a while, he’d just given up on pleasing the teacher. There was plenty of work to do out on the ranch and he didn’t get graded on it except by the cattle, and they wound up on somebody’s grill, so that didn’t matter.

“I do remember a joke though,” he said. “Wanna hear it?”

She looked like she wasn’t too sure, but he went ahead anyway.

“It’s about an Indian chief named Pythagoras,” he said. “He had three wives. One was big and fat and sat on a hippopotamus hide. The other two sat an the other hides. He said that the squaw on the hippopotamus was the sum of the squaws on the other two hides.”

That got the attention of his wife who’d been in the kitchen.

“Don’t go telling her stuff like that, or you will get her into trouble,” came a rather stern feminine voice from the other kitchen.

Jim smiled at his daughter. She smiled back and turned back to her books.

“I have to go out to work on the fence,” Jim said. “You help your maw. ”

With that the girl brightened somewhat and began to get up.

“Can I come too?” she asked.

“Nope,” Jim said. “You stay hear and work on your lessons and help your maw. You don’t want to be out today no how.”

His wife came quietly to his side and whispered to him.

“You know,” she said, “she’s not going to be wanting to be around you forever. Before long she’s going to get interested in boys and you won’t see her unless she has her hand out for a twenty-dollar bill.” Having finished, she quietly stepped back into the kitchen.

Jim screwed his mouth around a second or two as he thought.

“Come on, Punkin,” he said. “I just remembered a job I have for you.”

With that, Jim headed to the mud room, as it was called, and pulled on his boots. Then he pulled on his jacket and hat. Thus prepared he stepped outside and was greeted by a northerly breeze. His daughter was there at his side.

He faced the breeze and scowled. His own father had always said on days like this that there wasn’t anything between them and the north pole but a barb-wire fence and a couple of jack rabbits. He didn’t know about the jack rabbits, but there was going to be another barb-wire fence before it was over.

He climbed into his truck, his daughter hopping in to ride shotgun. He cranked the engine to life and headed down the dirt road. He’d purchased some more land a few miles away and he’d putting some improvements in on it. He’d rented a dozer and dug a couple of ponds for his cattle to drink from and now he was in the process of fencing it off.

“Well,” he said as he pulled off to the side of the road, “here we are.”

They extracted themselves from the truck and slammed the doors.

“What are we going to do, Daddy?” Kristin asked.

“We are going to so some practical geometry,” he said.

She looked confused.

“Building fence,” he clarified.

They began extracting the things they needed from the pickup’s bed and setting it up on the ground. The tools included three log chains of differing lengths, two piece of re-bar, and a mallet.

“You see,” he said pointing to the fence that ran parallel to the road. “I already have it fenced in. What I want to do now is to run a fence along the middle so I can have my cattle graze one side while the other side rests. I want the fence along the middle to be at an exact right angle.”

“Is that what the log chains are for?” Kristin asked.

He smiled. She was a sharp one, she was.

“That’s right,” he said. “The short one is three yards long, the middle one is four yards, and the long one is five yards.”

He handed her one end of the short chain while he took the other.

“Let’s run this along the fence that’s already there.”

They walked over and lay it along the ground underneath the fence. Once it was stretched out, he hammered re-bar through the last link on each end and into the ground, leaving about four inches sticking up.

“Run and get me that middle-sized piece,” he said.

Kristin gamely ran over and picked up the piece of change that was four yards long and carried it back over. Jim saw it was heavy, but as she didn’t ask for help, he didn’t give it.

Before he could say anything else, she ran over and drug the 5-yard piece back. Only someone who’d known him as long as his wife had would’ve been able to see the smile on his lips, and she wasn’t there.

“Okay,” he said, “you put the last link in that piece over the re-bar on your side”--he pointed--”and I’ll do the same with mine. But first we need to crawl over to the other side of the fence.”

After they’d crawled on their bellies to the other side, they did as he said, and then he took the free end of her chain.

“I am going to walk these taut,” he said, “and you make sure the other ends don’t come loose.”

He then walked into the field until the chains were tight and laid them all out on the ground.

“Come here,” he said, gesturing to her. “That’s a right angle. I am gone to put in my first fence post where the 4-yard chain meets the 5-yard chain. Since you can crawl under the fence better than your old daddy, run and get me my diggers and then start handing fence posts over.”

They then went to work, but as the business of digging and setting posts goes more slowly than the carrying thereof, Kristin soon finished and took to drawing in the dirt in a place where the grass was worn through.

After a while, Jim began to feel hunger pains and noticed that the sun was getting low.

“So, Kristin,” he said, putting his post-hole diggers aside, “do you reckon your momma’s got us some supper ready yet?”

Kristin, who’d been rather intent on whatever she’d been drawing in the dirt, had to forcibly extract herself from what she was doing.

“I guess so,” she said. “But, before we go, can we try something?”

“Sure, Sweetie,” he said. “What do you want to do?”

“Do you have a tape-measure?” she asked.

He nodded.

“It’s in the back of the truck,” he said.

Before he could ask why, she’d already crawled under the fence, fetched the tape measure, and brought it back.

“Hold that end at the corner post,” she said. She handed him the free end of the tape measure. She then walked down the fence that was parallel to the road.

“Three yards, four yards, five yards,” she counted. The she took a ribbon from her hair and tied it to the fence.

“Okay,” she said, “stay where you are.”

Jim was a bit confused, but like any proud father spending time with his daughter, he indulged her. Kristin, for her part, walked to the other fence and began counting out loud again.

“Six yards, eight yard, ten yards, twelve yards,” she said. As she uttered her last words, she found herself exactly by a fence post. “Okay, Daddy, you come her and hold the spool.”

“What do you want me to do with this end?” he asked. This was getting complicated.

“Leave it there,” she said. “We’ll switch placed.”

After they swapped, she began walking toward the piece of ribbon she’d tied to the fence earlier.

“If I am right,” she said, “that will read exactly thirteen yards when I get to the ribbon. I want you to check for me.”

It didn’t seem likely to Jim that she would walk five yards down the fence and it would only pull out one more yard of tape, but he kept it to himself. Kids have to try things.

He watched the tape come out. It was too fast to read, but then she stopped.

“What does it say, Daddy?”

He looked and uttered a profanity.

“Daddy?”

“Well, I’ll be John Brown,” he said. “It’s dead on thirteen yards. How did you do that?”

She smiled a smile that warmed his heart.

“It worked with 3, 4, and 5, so I wondered if it worked for any other numbers, so I figured it out on the ground.” She indicated the place in the ground where she’d been scratching. There were equations there that read: a=n2-m2, b=2nm, and c=n2+m2.

“You can do that for about as many numbers as you want,” she said. “What do you think, Daddy?”

“I think that me and your momma’s gonna have a long talk after supper.”

Summing it up

2011-06-23T18:50:00.001-07:00

Summing it up

By Bobby Neal Winters

Being a mathematician is a special thing. We mathematicians have been touched by God--touched at least--but it’s not without its cost. Part of that cost is being set apart as a separate branch of humanity, considered somewhat strange by one’s fellow man. Another is spending a large part of your life distracted by one problem or another.

This was the case with me the other day as I was coming home from a neighboring state. I was set apart because I was driving alone, but mainly I was distracted by a problem. The maddening thing is that now I can’t even remember what the problem was. This is another one of those things which sets us apart from humanity, a certain forgetfulness in the simple matters of living.

In any case, I was distracted for some extended interval of time, and, as happens so often when we go off on these little mental vacations, the rest of the world keeps going on around us. In this particular instance, my car kept finding its way along the road, but it didn’t keep going along the right road. I missed a turn. When I shook off my distraction, I was off the main highway and in the heart of what we in these parts, in homage to Ned Beatty, refer to as banjo country.

I came to a place where the road widened slightly and to the right side there was an unpainted wooden building with a sign above the door that read “Groceries.” There was also a sign that said “Cold Pop.” As it was a warmish day and as I had been riding along with the windows rolled down, the idea of a cold pop appealed to me.

I stepped out of the car onto the gravel and walked to the store. I couldn’t tell whether the store had ever been painted, but the door, which looked to have been a later addition, had been green at one time. It was a wooden frame with rusted screen wire held to it in a rather haphazard fashion by tacks. When I opened it, it screeched so loudly that the bell above it seemed superfluous.

This was all misleading, however, because when I got inside everything seemed new. And as I reread that last sentence, I realize it is misleading because it wasn’t new, it was old, but looked new. The pop machine was an antique, an expertly restored antique, but an antique none the less. I looked inside and it had been filled with nostalgia brands of soda pop which aren’t available anymore along with currently popular brands in classic packaging. Before I could extract the pop which and lured me into the store in the first place, I was betrayed by the mathematician in me and became distracted by the rest of the place.

The food shelves had been restored in a fashion similar to the pop machine and were stocked with products in old-style containers. My eyes ran along them and were drawn immediately to the coffee which was in old-fashioned cans. I have sorely missed the metal cans that coffee was sold in not so long ago. My first thought was I wanted to buy some. My second thought was that in these fancy, classic cans that it would be very expensive. My third thought was that it wouldn’t hurt to simply see what the price was.

My fourth thought was complete astonishment. The cans were fairly small but their price was a fraction of what I usually pay. It don’t remember exactly because the price is not important, but let’s say it was $1.22 for a size I usually pay $5 for. I must have said “Wow” to myself or something because I heard a voice from the direction of the counter.

“Kin I hep ya, Mister?”

It was an old fellow who was clearly in costume. He wore a crumpled old felt hat and had a corncob pipe jammed between the jaw teeth on the right side of his mouth.

“Uh...yeah,” I stuttered, “are these prices real?” I asked.

“Yup,” came the answer.

“Is there any limit on the number of cans I can buy?” I asked again, not daring to hope.

“Nope,” was the single syllable reply.

I began to load up my arms. I don’t know how I managed it, but I got thirteen cans total and, forgetting about the soda pop I’d come in for completely, toted them up to the front.

I sat them down on the counter and he eyed them.

“Thirteen?” he asked.

“Thirteen,” I confirmed.

It was then I noticed that he had several cans in front of him containing different objects. From two of these he removed numbered disks: one red disk with $1 on it, two green disks each with 10 on them, and two blue disks each with 1 on them. He laid these out in a row in front of himself and to the right.

He then reached into a can and extracted some black objects that looked like carved scarab beetles. He laid them out into two rows: one six scarabs long and the other seven, making a total of thirteen.

What happened next happened with a speed that only comes as a result of practice. I will try my best to explain it to you. One of the rows of scarab beetles was one longer than the other. He removed the extra scarab from it and set it my his row of disks. He then picked up an entire row of scarabs and put them back into their can.

Then he took more of the colored disks from cans and laid out exactly twice as many of each of the different kinds as he had before.

Then he laid out the remaining six scarabs, left over from the first round, into two columns. Actually, a better way to describe it would be to say he put them into three rows of two. This was so that he would not have to do any mental activity at all; he just laid out the scarabs mechanically. As there wasn’t an extra one left over this time, he didn’t set it by the row of disks, but he did take one of the columns and put it back in the scarab can.

He then took numbered disks and doubled the number he’d put down in the previous round, and, as before, laid out his scarabs. This time there was one row of two and one left over which he put by his pile and put one of the remaining two back in the scarab can.

I could see this process was going to soon stop so I watched carefully. He doubled the numbered disks one last time and put the lone remaining scarab down next to it pro forma.

Then he put away the numbered disks from the rows without scarabs, counted the remaining ones, and said, “Ya owe me fifteen dollars and eighty-six cents.”

I did a little mental guestimation--as I am lousy at arithmetic--decided it was a fair price in any case, took out the money, and paid the man.

I carried the coffee back out the the car, loaded it and myself up, and drove off. On the way home, again distracted by what I had seen, it came upon me what the old coot had done. It is know as the Russian Peasant Algorithm or Egyptian Arithmetic, depending upon whether you are trying to impress Russian peasants or Ancient Egyptians.

It is a mechanical process which will literally produce the number of objects needed. In societies that had not yet discovered a way to represent numbers that makes arithmetic convenient, a process such as this one would be quite valuable.

Lets consider how one could multiply seventeen by thirteen. It is much more fun to do with tokens, of course, but in writing it is more convenient to do it as follows:

17	13 *
8	26
4	52
2	104
1	208 *

Note I’ve used * in place of the scarabs. The sum of the starred number is 221 which is the product of seventeen and thirteen.

Those of a mathematical bent will, of course, be led to think of binary numbers. One should be aware that the Egyptians where doing this when all the silicon chips in our computers were still sand.

When I ran through all of the cheap coffee I bought that day, I made my way back over to the neighboring state and tried to find that dilapidated old store, but I wasn’t able to.

Decision time

2011-06-06T17:54:00.000-07:00

Decision time

By Bobby Neal Winters

An old friend of mine rolled into town the other day and we headed over to one of the local upscale coffee shops to get a designer coffee. There are a lot more decisions involved in getting coffee now than there were in the days of old. I’ve subverted the system by simply getting the same thing every time. It is a beverage I like to call “coffee.” The barista knows what I want when I asked for it, and I can then put cream and sugar into it--or not as the availability dictates.

My friend is cut from a different bolt of cloth, and he explores permutations making decisions according to what he’s tried before and giving preference to the new and different over the old.

“I like to try new things,” he told me after putting his lips to his latest concoction. When he sat down his cup, he was left with a whipped-cream mustache that he wiped away with a napkin.

“I am glad,” I said, sipping at my tried and true, no-nonsense coffee. “You can explore for the rest of us.”

My friend, who has been in the business world since graduation, looked sort of wistful.

“You know,” he said, “I sometimes wish that I had the sort of freedom to explore new ideas in my work that you guys do at the university. In business, most of our decisions are determined by policy and there is not much incentive for exploration.”

He’d caught my attention when he said this, because I teach hypothesis testing as a part of my Elementary Statistics course, and I am always looking for real-life examples.

Hypothesis Testing deals with creating schemes to make decisions. The hypotheses that the students are given are quantitative statements that are either true or false but which, by their nature, you can’t ever be certain. You can’t know whether its true or false but you set a procedure to decide whether or not to treat the statement as true or false.

I explained all of this to my friend while contemplating the possibility of a second cup of coffee and maybe a muffin to go along with it. I was hoping to be able to get my friend’s opinion of whether the sort of examples I use have any sort of connection with reality. He’d opened his mouth and was about to say something when I heard a voice coming over my shoulder.

“Yes-sir-ee, sometimes decisions are hard.”

The voice had a pronounced Texas; the word “decision” was pronounce DEE-cis-zhun. The voice was attached to a person whose appearance wouldn’t lead one to expect anything else. From the wrinkles in his face, I judged his age to be anywhere from 70 to 170. He wore a new, clean cowboy hat made out of straw, a spotless white shirt, and what I took to be a brand new pair of blue jeans. In Texas, Oklahoma, and similar places, there is a subculture in which such an outfit will work for church, doing serious business, or, apparently, getting a designer coffee. How he’d made it to this part of Kansas, I couldn’t guess.

I was going to just smile and ignore him, but I guess he caught my friend’s interest.

“So you’ve made some hard decisions?” he asked.

“Well,” the old man drawled, “I guess I have. I guess that anyone who’s lived as long as I have would’ve made some hard decisions, some good, some bad. I ain’t never had a class on it though. I was just thinking about the first time I ever rode in an airplane. It was almost the last.”

Usually, I am a good listener, but I couldn’t see where this connected and I was more interested in listening to what my friend had to say than whatever this rustic, old fellow had to say, so I attempted you re-engage my previous conversation. I turned to my friend, looked him in the eye and spoke.

“You were saying...”

My friend’s attention had been captured, however.

“Oh, I wasn’t going to say anything important,” he said. “Just some business stuff. Let’s listen to what this gentleman has to say.”

So we listened.

I’d been following the wheat harvest from west Texas up though western Oklahoma, Kansas, and Nebraska. I’d gotten tired of all logging all of those miles back to visit this girl I was trying to spark back home, so I thought that I’d want to learn to fly one of those airplanes.

But I figured I wanted to ride in one first before I flew it. It just sort of made sense to me. So I was talking about it with a fellow by the name of Jack “Hot Shot” Burns. Jack was a good old boy. He’d spent some time out west fighting forest fires in the mountains. He’d fly his plane out over the fire, drop a big load of water, and put it out.

It’d kind of got to him though. He was as fearless as any pilot that ever was, except when it came to fire. When I was talking to him, I lit up a cigarette, and he about come unglued. I had to stamp it out right away and put my lighter out of sight.

Anyway, Jack and me and Floyd Jumper was talking over beer and I said that I wanted to learn to fly, but I figured I’d better go up in a plane first. And they said it was a good idea.

Floyd had been a parachute jumper in the Army and he and Jack had a thing going where they’d take people up and teach them how to jump.

“I don’t want to learn how to jump,” I said. “Just how to fly.”

They said that they wouldn’t make me jump, but, as they were doing that all the time anyway, it would be easy to take me up. So we finished our beers, went out to the airstrip, and took off.

Now, you fellows probably are used to those airports like they have in Kansas City or Tulsa. This wasn’t like that. It was an airstrip out in the west Texas plains. There wasn’t anything but wheat or grass or cattle as far as the eye could see. We got into his airplane which was barely big enough for the three of us, I tell you.

I could see that it would be a good plane to parachute jump from because I wanted out of it from the minute I got in. Jack sat in the pilot seat of course; he had Floyd sit in the back; and he had me sit up in the seat usually reserved for the co-pilot so that I could see better what flying was like.

Then we look off.

It was something. After we climbed to about 10,000 feet, I got over the pure terror of being up in the air without anything but a thin piece of tin between me and eternity and I got to thinking that maybe I would like to fly one of these machines, so I started looking over the console.

There were a lot of gauges, nobs, and lights on the dashboard and it struck me that it would be a lot to keep up with, but I figured that was something you’d be able to learn. It all looked good to me but I saw a red light on, so I figured I’d ask a question.

“Jack,” I said. “What does this light mean that says ‘FIRE WARNING.’”

Jack didn’t answer. He just got a crazy look in his face and went berserk. I’ve never seen a man so panicked. He tried to climb out of the plane over me, and when I turned him back, he opened the door on his side of the plane and just stepped out. No parachute, no nothing. Jack was gone.

Floyd hopped into the front seat and closed the pilot’s door. He then leaned over and looked at the light. He called me a name I can’t say in front of polite people like yourselves. Then he tapped the light.

“That says ‘TIRE WARNING.’” And he called me the name again.

I couldn’t see that this was my fault because Jack had made a poor decision. I don’t know about you, but it is going to take more than one person’s word to get me out of an airplane without a parachute.

There was a period of awkward silence when Floyd appeared to be doing a lot of thinking. I couldn’t take much of that so I asked a question.

“So, I guess you’ll be landing the plane now so we can tell them about poor old Jack.”

Floyd looked at me pretty serious then.

“I don’t guess we can do that,” he said. “You see, I don’t know how to fly a plane either.”

Since then, I’ve heard stories about folks being in situations similar to ours and having people on the radio talk them down. We hadn’t heard of any of that, and we didn’t think about it because the radio in the airplane didn’t actually work.

Floyd finally got figured out whatever he was thinking about, and he turned to me.

“Fella,” he said, “I don’t know how to fly, but I do know how to jump and I can teach you. Usually, we have some classes on the ground first, but now there’s not time. You can either learn or crash with the plane, but I am jumping before we get too much farther from the airstrip.”

Now you’d be surprised how fast you can learn when you have to. He talked me through all of the procedures, and then we were ready.

“We’ll jump at the same time,” he said. “You count to ten and pull. When I see you are okay, then I will pull.”

Jumping out of an airplane is hard even when you know it’s going to crash, but I did it. And I counted to ten. I actually screamed my numbers. Then I pulled and that parachute opened with a jerk.

Floyd looked, saw that I was okay, and even gave a little nod. Then he pulled his rip cord.

And nothing happened.

Again, being it’s such polite company, I won’t tell what I saw next, but I will say I won’t ever forget it.

“So I guess the moral to this story,” he said with a grin, “is that when the stakes are high unless you’ve got pretty good you’d better stand pat. Jack didn’t follow that rule. If there is another moral, it’s that sometimes you can make the best decision possible and still wind up flat. That’s what Floyd found out.”

My friend was totally intrigued.

“What did you find out?” he asked.

“Nothing beats luck,” the old-timer said. “I went to the place where poor Floyd landed and found oil. That was the first of many wells. I am a billionaire today.”

Then he left. I looked at my friend and shook my head.

“The decision I have to make is whether to believe a single word he said.”

I heard another voice. This time it was the barista.

“I’d believe him if I was you,” she said, holding something in her hand. “He just tipped he with a Canadian Maple Leaf.”

Cora Layton

2011-05-18T16:07:00.000-07:00

Cora Layton

By Bobby Neal Winters

Math folks, like everyone else, enjoy congregating with their own kind from time to time. By own kind, I mean to say people of similar, if somewhat strange, interests. We do this at conferences. Over the years, I’ve attended conferences featuring talks in a variety of mathematical and mathematically-related topics. Aside from the talks, the real purpose of these conferences is to establish human relationships between real, living human beings.

One problem with this is that it is hard for anyone to envision those of us who practice the arcane mathematical arts as human beings real, living, or any other way. Nevertheless, I have, over the years, managed to get to know a few of my fellow math-types. One of these is Cora Layton.

Cora worked for many years at Sutherland State University teaching statistics. Statistics is not as some people think--including some mathematicians--a branch of mathematics. It is a proud discipline of its own that uses mathematics.

Cora had not only taught students, but she’d also done considerable work helping her colleagues in the sciences and social sciences with some of their statistical problems.

At the last conference, I’d heard she’d retired. Imagine my surprise at going to a mathematics meeting, looking up, and seeing her there.

“Cora,” I said. “Is that you?”

Her smile let me know that indeed it was her.

“Well, who else would it be?” she replied coyly, her southern accent so thick and sweet that you could imagine honey dripping from it.

“I’d been told that you had retired,” I said.

“Can we ever really retire?” she asked. I’ve heard this question asked a lot, mostly in relation to retirement accounts depleted by the stock market crash.

We left the conference and went for a coffee so that we could catch up with each other’s news.

As we sat there with our four-dollar designer coffees, I discovered that she had, in fact, retired from teaching. However, she’d needed to keep busy and hung out her shingle as a consultant. Doing this, she was able to stay only a bit busier than she wanted to be.

I said she’d hung out her shingle. These days a shingle need only consist of a page on the Internet. Being a teacher, I am always on the look for real-life applications for the classes I teach. I thought that in her new role she would be able to share some.

“So, Cora,” I began my inquiry, “have you run into any interesting problems that I could share with my students?”

Cora smiled.

“I don’t know that I’d call this a real-life problem,” she said, “but I found it interesting.”

She took a sip of her coffee and related her tale.

At first, most of her work had been a continuation of her role in helping her former colleagues at Sutherland State with their projects. Then one day she’d been contacted “out of the blue” as they say by a couple of men in their late twenties or early thirties.

The first contact had been by e-mail and had looked official enough. They had given their names as William R. [redacted] and James J. [also redacted] and bottom of the e-mail boldly claimed them to be representatives of the Weather and Atmospheric Cloud Organization, WACO for short.

Cora was to learn that they were more widely known as Billie Bob and Jimmie Joe. She’d smiled to herself when she read the acronym WACO, figured that they probably hadn’t called in a PR guy on the project, and thought no more about it until she met them. When she met them she thought the acronym was quite descriptive.

They drove up in a brand new Dodge Ram truck. It had a shiny, red finish and an even shinier chrome roll-bar. Sitting in the back was some rather expensive-looking equipment and attached to the roll-bar there was rotating satellite dish.

Billie Bob and Jimmie Joe are tornado chasers. While there are a good number of respected scientist who chase tornadoes, Billie Bob and Jimmie Joe are not among these. They consider themselves scientists; they’ve printed up letter head and t-shirts for their WACO organization; they take pictures and have all sorts of fancy equipment; but they fall short in the area of credentials and any sort of grasp of the scientific method.

To their credit, having seen other, more reputable tornado chasers in their activities, they’d come to the decision they wanted to pursue their passion in a more serious way. For this reason they had come to Cora wanting help on a scientific project.

The idea was this. They had observed in their various storm chases the devastation that tornadoes wreck on mobile homes. They are not unique in this observation and had began to wonder whether there was something about mobile home that attracted tornadoes. They had seen Cora’s page out on the Internet where she helped scientists, and they were scientists that needed her help.

“I almost sent them away,” she told me there at the coffee shop. “I don’t know whether it was because they were about the same age as my grandsons or whether they offered me $1000 up front and I remembered that saying about a fool and his money. In any case, I am a teacher and I can’t help but teach.”

Cora told them that they needed to collect data. She said that they needed to visit mobile homes, those which had been hit by tornadoes as well as those that had not. They should look for things that might reasonably be factors in attracting tornadoes.

She also introduced them to some simple statistical software and had instructed them in the use of it.

“I figured that I ought to be doing something useful for the money,” she said, looking only slightly guilty.

It was much to her surprise when they returned a few weeks later with data.

“They had taken by advice very seriously,” she said.

Billie Bob had noted the presence of car batteries in the yards of many mobile homes. He’d figured that, as they could be used to produce electricity, it was conceivable this might have some effect on tornado-producing storms. Jimmie Joe had noted the presence of old tires mounted on rusty wheels in the yards of many of these mobile homes. He’d wondered whether the roundness of the tires might have anything to do with attracting tornadoes as tornadoes turned in a circle.

They’d made careful counts of these at numerous trailer parks and fed the data into the computer. What they’d found had astounded them.

“They didn’t find any significant difference in the groups that had been hit by a tornado and the control group,” she said with her lips betraying just a hint of a mischievous smile. “What they did find and what did astound them was a correlation between the number of tires and the number of batteries. The number of tires is approximately 3.95 the number of car batteries they found. The 3.95 is accurate plus or minus a margin of error of 0.1.”

I thought about this myself a minute and began to share her smile.

“Did they, by any chance, in the course of collecting data, note the number of old cars up on blocks with their hoods open in these yards?” I asked.

She smiled even more broadly.

“They did not,” she said. “But I did suggest they broaden their study to include the width of satellite dishes and sofas on the front porch. My sister lives in Florida and it’s been a while since I’ve seen her. If they do a little more science, I might be able to afford the trip.”

Marge Anovarra

2011-04-26T14:16:00.000-07:00

Marge Anovarra

By Bobby Neal Winters

“Hey, Doc,” I heard the voice from across the restaurant. I thought the voice was familiar, but as I looked up from my barbecue ribs to the direction whence it came, I didn’t recognize it. I must have looked confused because I heard the voice again, and this time I saw who was speaking.

“Hey, don’t you recognize me?” the woman in her mid-thirties said. “I must have really gotten old.”

It is always so cute to hear 35-year-olds say that. But after subtracting a minor amount of gray from her hair and an inch or two of width from her face, I did know her.

One of the great joys that a teacher has is keeping in contact with former students. Not all of them, you understand, but certain ones do bring us a great deal of joy. Such is the case with my former student Marge Anovarra.

Marge was not a math major. She was a Communication major from Krebs, Oklahoma. She was a proud Italian and a proud Okie. Pride isn’t rare in either of those races, but it is rare to see the Italian and the Okie combined. The reason it is combined in her case is the same reason that she chose to come to my university: mining.

It is little known that there has been coal mining in both Southeastern Kansas and Eastern Oklahoma. The Italians immigrated to both places to take part in that. Marge came to my university because it allowed her to take part in her Italian cultural it a place that was not home but like home and not terribly far from home. And, as one might expect, after she finished her university degree, she moved back home.

She now practices her talents as a journalist and free-lance writer throughout the region and is successful.

She first caught my attention in class when I was calling roll on the first day. With a name like Anovarra, I assumed she was a local and started joking with her. I told her that we would learn more about her during class when we started talking about confidence intervals and the margin of error. The class laughed, but afterwards she approached me an explained in very serious tones.

I was informed that her last name had been changed when her ancestors came to this country. The folks who’d registered them had been in a hurry and when her great grandfather was asked his last name, he’d stuttered.

“Uh…Novara.”

Thus the last name preserved that error from that time onward.

When we did get to margin of error, Marge—and that was her first name, not Margaret—did like it. She liked the entire practical turn of statistics. It’s based on the idea that even if you can’t know something for sure that shouldn’t keep you from trying your best.

The concept of margin of error first occurs in the calculation of confidence intervals of the mean. The idea is that you want to know the average of some number associated to a group of objects, such as the average shoe size of everyone on our campus. This is not information that our admissions people ask for, so if we really want to know this, we would either

1) phone everyone on campus and ask, “What’s your shoe size?” and listen to approximately 7000 phones slam in our ears, or

2) take a sample and use the sample to find an estimate.

The latter course is simpler and less expensive. The idea is that a random sample of a group—a population being the technical term—models that group. The average on that sample should be close to the average of the whole group. There are ways this could go badly wrong. You could choose only to sample from the men’s basketball team who would have larger feet than the typical folks on campus; or you could choose to sample only the female gymnasts, who would tend to be smaller than typical. But the idea is that a randomly chosen sample will, for the most part, yield an average closer to that of the group.

It needs to be noted that the average of the sample will tend to be close to the average of the population, but it will almost certainly be wrong. It is for this reason that statisticians give themselves some leeway: the Margin of Error.

So you start with the average of the sample and you know that you are wrong. Your average might be too big, so you subtract the margin of error from the sample average; your average might be too small, you add the margin of error to the average of the sample. After all this, you still might be wrong, but the probability that you are correct is called the level of confidence.

You calculate the margin of error with this level of confidence in mind. So estimating the average shoe size of campus won’t give you a single number. It will give you an interval of numbers and you still have a small chance of being wrong.

That appealed to Marge’s practical nature, and it all came back to me in a flash, a microscopic fraction of the time it has taken me to relate it to the reader.

“Marge,” I said, not disguising the joy I felt in seeing her after an interval of 15 or 20 years. (I have to leave myself a margin of error you understand.) I invited her to join our table when we were sampling not only ribs but brisket and pulled port.

“It’s great to see you,” I said after she placed her order. “I understand you are a journalist now.”

“That’s right,” she said. “I write for a few papers and do some freelance work. Right now I am working on a piece for the Morning Ritual. The Morning Ritual is the house publication of a health food company that encourages the consumption of bran. The company is a sole proprietorship, owned by a militant vegan.”

After she ordered burnt ends, she then went on to explain that she’d been approached to do an expose of the barbeque restaurants in the region. The editor had approached her with shocking news.

“Almost half of the barbecue places in the region serve road kill,” he’d said in serious tones.

The research she’d done in preparation for her article didn’t bear that out. Out of a sample of 143 restaurateurs, 51 of them said that they had served road kill. She’d done the calculation and that had only worked out to 35.7 percent.

“That seems awfully high, but it’s not close to half,” she said. “I remember you teaching about margin of error, but I’ve forgotten how to apply it here.”

My family looked strangely at their plates.

I, on the other hand, took out my pen and did a little figuring on the napkin. Then I called the waiter over to borrow a calculator. I figured that the margin of error was 7.8 percent. This meant that the fraction of restaurants serving road kill was between 27.9 and 43.5 percent with 95 percent confidence.

“What might’ve happened,” I said, “was that someone looked only at that upper part of the estimate and rounded it to 44 and from there it’s not so hard to say almost half.”

This isn’t a very ethical way to present that particular data, but looking at the other end of the confidence interval is awfully disturbing too. It is saying that over one-fourth of the barbecue restaurants serve road-kill.

“Where was this survey taken?” I asked.

“It was at a barbecue festival,” she said. “But I see where you are going. I will do a little more digging to find the particulars. Thanks.”

She then finished up her burnt ends and excused herself. I gave her my business card with my e-mail address on it as she left, asking her to keep me filled in. It was a week before I got an e-mail from her.

It turns out that it wasn’t clear these people were restaurant owners at all. They were simply people that attended a barbecue festival catering to people who cooked wild game. The survey was administered at a workshop entitled “Using a dark sauce to cover tire tracks.”

“That kind of explains it,” she wrote. “I wasn’t able to use the data, but digging to get to the bottom opened up a whole new subculture that I was able to write about.”

Custer’s Last Stand

2011-04-21T14:49:00.000-07:00

Custer’s Last Stand

By Bobby Neal Winters

We are told that a prophet is not without honor except in his own home town. And it’s true that those of us who know us best have the most trouble believing that we might know anything ourselves. This is especially true, in my opinion, when you practice mathematics. I think this is because that it can be so technical that non-professionals can’t tell what is relatively easy from what is hard because they don’t understand either.

At times it has been important to me to convince my kinfolks that I do know something about mathematics. This is made difficult for me because I am not very good with arithmetic. If you ask me to add up a column of figures, I do like everyone else and reach for a calculator. Add to this the fact that my expertise is in the topology of non-compact three-manifolds which is something that doesn’t just naturally come up in conversation after Thanksgiving dinner.

I have learned a tiny bit of statistics by teaching it over the years and that has provided an opportunity to help at least one of my kinfolks recently. That kinsman was my cousin Custer.

Custer is the grandson of my Aunt Vidalia. He was her special pet when he was growing up and she spoiled him rotten. His parents spoiled him rotten as well. They are fun to spoil, but sooner or later there is a price to pay, and Custer has had a problem that in some circles is referred to as failure to launch.

That is to say that they can’t get him to move out of the house; he just hangs around and obsesses on Bigfoot all day.

The reason for this has to do with work. It’s not that he can’t get a job. He’s had lots of jobs. Vidalia has lots of friends who have either given him jobs or found jobs for him. It’s keeping the job that’s the problem. Some of the elders in the family have been of the opinion that if he spent half as much time working as Vidalia’s friends have in finding work for him that he would be president of Standard Oil by now, but I believe in giving everyone a chance.

His latest job had been at the Sucker-Rod Museum. According to its brochure, it was the home of the world’s largest pile of sucker-rods. For those of you who didn’t grow up in the oil field, a sucker-rod is a cylindrical piece of metal. They are about an inch in diameter and about 30 feet long. None of this is important to what comes next, but I didn’t want you to be worried about not knowing that.

The job had been given to him by one of Vidalia’s friends who is big in the oil field. He’d set up the Sucker-Rod Museum and had needed someone sit at the reception desk, invite the visitors to sign the book, and to keep track of the number of folks who visited. Previously this job had been given to a retired old-field worker who was confined to a wheel chair. The job had opened up when the old fellow had died at the age of 92.

The fellow had taken the job seriously and had kept an immaculate set of records. I know this because I’d seen them as a part of a project I’d done for Custer.

Not long after getting the job, Custer had e-mailed me with a project he was working on. He had the records that his predecessor had collected and wondered if there were ways of predicting the number of visitors to the Museum on a given day.

Now, I get nervous when people start using that word “predict.” We can make statements about trends over time given certain assumptions, but we can’t actually be prophets. People want magic, however, and sometimes it’s hard not to get sucked in.

I answered his e-mail and told him to send me a copy of his records and I would see what I could do.

The Museum is opened 6 days a week, Tuesday through Sunday, 52 weeks a year. It’s closed on Christmas and Easter, so that usually works out to being open 310 days a year. The last year that I had full data on, it had 300 visitors. This was an average of 0.97 visitors a day. The other years were off from this slightly but not by much.

The idea of having an average of 0.97 visitors can be somewhat comical, as we try to imagine what 0.97 of a person is. Would these be old oil-field workers missing a few fingers or a hand? Maybe that’s not so comical. Anyway, that is not how averages work, of course. That this means is that some days several people would come in and some days nobody would be there. While on any given day, it is not necessarily predictably what will happen, one can sometimes say something about the distribution. By this I mean to say that there will be a certain fraction of the days when no one will come in, there will be a certain fraction of the days with one person will come in, a certain fraction when two will come in, and so forth.

Often situations like this will follow the Poisson Distribution, which is also known as the Law of Rare Events. This is kind of misleading because the Law of Rare Events can also model very busy places like restaurants and barbershops. The word rare is a reference to on any given day the probably of a particular person going to a particular restaurant might be small. The distribution depends upon the average number of visits per day, such as the 0.97 for the Sucker-Rod Museum.

I examined all the data he sent me, and it turned out that the pattern fit the Poisson Distribution very nicely. I sent him a report saying that on 38 percent of the days there would be no visitors, on 37 percent of the days there would be one visitor, on eighteen percent there would be two visitors, on six percent there would be three, and on one percent there would be four.

I swear that is all that I told him.

A few weeks later I got a hot call from Custer’s father, in whose basement Custer is living, telling me that I’d gotten Custer fired.

“What?” I asked, hurt and confused.

“He said that you’d done some math and he used it and was fired because he used it.”

After saying that I didn’t see how what I told him could get him fired, I said that I was sorry that he’d lost his job, then I hung up, and made a few phone calls. After talking to several people, the following picture emerged.

Custer observed that 93 percent of the days there would be two or fewer visitors. He then deduced—correctly—that only on seven percent of the days were there more than visitors. He then developed the habit of closing the museum after the second visitor left deducing—incorrectly—that where was only a seven percent chance that anyone else would come.

Over the course of time there were complaints from people who showed up to the museum wanting to see the world’s largest pile of sucker-rods only to find the museum closed.

My final call was to my Aunt Vidalia. Who explained how it came to an end.

“Custer has always been a big fan of Sasquatch. It all came crashing down on the weekend of the Bigfoot Festival in Honobia, Oklahoma,” she pronounced Honobia as Hoe-nubby just as its inhabitants do. “He’d taken that weekend off without telling his boss. He was afraid he wouldn’t get paid if he told him. Then the boss came by. Well, he just didn’t understand.”

I commiserated a bit with Vidalia over this before I said good-bye.

Custer made quite a few mistakes here. The non-mathematical ones I leave to you. The mathematical ones are a bit subtle and don’t have anything to do with the Poisson Distribution which I’d done correctly.

Custer’s problem was his assumption that if he left after the third visitor there was only a seven percent chance that there would be another one. It turns out that if you’ve already had three visitors, then there is about a thirty percent chance of having more visitors. This is because already knowing that you have had three visitors changes the initial assumptions of the problem.

It’s called conditional probability and it catches a lot of people by surprise. The one thing that is not a surprise is that Custer is out of work, and the family is blaming it on me.

Keeping track of the factors

2011-04-14T12:24:00.000-07:00

Keeping track of the factors

By Bobby Neal Winters

You may recall my having told you about a person named Wilma B. Even in a piece entitled “Uncountable Candy.” Wilma is the president of the United Methodist Women in the First United Methodist Church in Kimberly, Kansas.

By the way, don’t look for Kimberly, Kansas on any map because it has so far avoided the attention of any cartographer. It was christened Kimberly after Kimberley, North Cape, South Africa, a place where diamonds were discovered. It’s in Kansas, there were never any diamonds discovered there, and the folks who named it left out the last “e” because it didn’t look to them like it was actually needed. They are a parsimonious lot in Kimberly.

In my mind, the name fits, however, because the people there are such jewels. Wilma is one of those jewels and personifies the spirit of the place better than anyone else I know.

The parsimony of Kimberly is a result of its location in what in many ways is still a frontier. As one looks west from Kimberly, except for the occasional town and small city, one sees a gradual thinning of population, a great emptiness. One gets the feeling there is no one around to help, and one learns not to count on help. This argues against waste. It argues against change.

Wilma is a product of this environment and one of its best exemplars. One of her characteristics is that she finds a way to do something and then falls into the trap that it is the only way or the best way. Such was the case during her church’s Vacation Bible School a few years ago.

It had been her church’s custom to give the children different gifts on each day of the week: on Monday they would receive crayons; on Tuesday coloring books; on Wednesday scissors; on Thursday glue; and on Friday they would receive glitter.

Looking at the list, there is a certain logic in it. I can envision a reality in which there would’ve been reasons associated with giving the gifts in this precise order. Indeed, I’ve been shown the curriculum the church used to use and there were needs for particular items on particular days. Due to the realities of the publishing industry, they’ve been forced to switch to a particular curriculum in which the children having their own scissors earlier in the week would be handy.

As with so many things that come my way by virtue of having a wide circle of acquaintances, I do not know exactly how everything occurred, and, what is more, I don’t want to know. I have been told there was an energetic discussion in which one of the Vacation Bible School teachers informed Wilma that there were a lot of different ways to do this. Wilma disagreed with that—energetically—and words were exchanged.

Let me emphasize that I knew nothing, and I mean nothing, of this when Wilma approached me. I am a friend of Wilma’s, and being Wilma’s friend has a certain amount of overhead associated with it.

We had met at an event—I don’t remember which one—in which donuts and coffee were being served. Before I had time to notice, she had laid out the situation involving giving out the gifts in a very neutral way and then proceeded cagily.

“How many ways are there of doing that?” she asked. “Would it take too long for you to figure that out?”

I had no idea I was being asked a controversial question.

“It wouldn’t take long at all,” I said. “There are 120 ways to do it.”

She seemed somewhat surprised with the rapidity of my answer.

“How did you figure that out so quickly?” she asked. “I am not sure I trust an answer that you can get so quickly.”

“Oh,” I said, “this is something that I teach almost every semester. When you make an ordered list of five items there are five factorial ways of putting those items in order. Five factorial is equal to 120.”

She was beginning to look a little grim.

“I am not sure what you are saying,” she said. “Five factorial?”

“That’s right,” I said. “Five factorial is five times four times three times two times one. That is equal to 120.”

“Okay,” she said, “I understand that five times four times three times two times one is equal to 120, but why is that the number of ways of putting five things in order.”

I’d just finished my donut and had a napkin in my hand that was no longer serving a purpose, so it was at that point I pulled my pen from my pocket and drew a diagram on the aforementioned napkin that looked something like this.

“For the sake of time,” I began, as I showed the napkin to her, “I’ve decided to consider the case only for the first three days. I am sure that this will be enough for you to see the pattern.”

We mathematicians are optimistic creatures and have confidence that in following a pattern three times is enough for anyone to get it. As with most optimists, we are constantly disappointed. Nevertheless, she nodded her head that she was willing to buy into the proposition.

“On the first day, there are three choices: crayons, books, and scissors.”

She nodded, so I continued.

“Suppose that we choose crayons as our first choice. This leaves only books and scissors to choose from.”

Again she nodded.

“Then suppose on the second day we choose books. This leaves scissors as the only thing left to choose on the third day. That path along the top in my graph represents that particular series of choices.”

She nodded again.

“Noticed that this graph I’ve drawn branches.”

Again there was a nod.

“This is because there are choices to be made. On the first day there are three alternatives, on the second day there are two, and on the third day there is only one choice. Notice that we each sequence of choices corresponds to one of the paths in the graph. The number of paths can be obtained by multiplying three by two by one. If we go the whole week, it is the same principle.”

She wasn’t looking too happy, but this isn’t too unusual in people I explain math to, so I am used to it, but she was still listening to me. That, I am not used to.

“You call this ‘factorial’?” she asked.

“Yes,” I said. “You may have seen it written out as an exclamation mark.” I turned over the napkin and wrote it out.

3!=6

5!=120

7!=5040

“See, they get quite big,” I said. “That takes a lot of people by surprise.”

“Yes, it does,” she said.

I didn’t learn until later the story behind it all.

The porcelain pig problem

2011-04-04T13:49:00.000-07:00

The porcelain pig problem

By Bobby Neal Winters

Ever since learning about Marcus Nathaniel Release, the mathematically talented twelve-year-old cousin of a friend of mine, I enquire about his progress. Talent in a particular area is one thing, but success is a mixture of factors. If a talent is to be used, it has to be developed to be sure, but there is even more. One cannot allow talent to be an excuse for poor behavior.

As you may recall, Mark had estimated the size of the large family next door with the use of a paintball gun. Rather than in receiving a punishment, this had resulted in the mother of that large family inviting Mark over to play with her children. I wasn’t sure whether that was sending the young man the right message, but, as I don’t even know the family, that is none of my business. And if I’d brought it up, you can be sure I would’ve been told so.

I do make inquiries about him through my friend because, among other reasons, I’d like to see him as a student at my university one day.

“How is Marcus?” I asked. “Has he painted any more of his neighbors?”

My friend, who’d just taken a bite of a doughnut, took a napkin to wipe the corners of his mouth. He then smiled a little, in a mysterious sort of way.

“No,” he said. “At least none who weren’t also shooting paint at him at the same time. He’s become good friends with that family next door. The mom has sort of taken him in as another son. You know how it is with large families; there is always room for one more. He eats supper with them about half the time and he’s even started going to Sunday School with them.”

I nodded. Having grown up in a similar locale, I am familiar with the culture and none of this sounded strange, though I did notice a bit of a mischievous smile playing at the corner of my friend’s lips, so I remarked upon it.

“You’re smiling,” I said. “Is there something funny?”

“Funny?” he said, smiling even more broadly. “To me and you, maybe. To those involved, it was deadly serious.”

In the Ozarks, there is a high value put on the native, homespun crafts. The crafts are everywhere. There are numerous local festivals. There are theme parks. And, as religion is one of the principal conduits of culture in the region, crafts are integrated into church activities.

The church to which Mark’s neighbors belong is rural, but, in spite of that, has a large congregation. It is big enough so that they can support an impressive physical plant that includes a gymnasium. Every year they open their gym to local artisans so that they can display their wares, sell a few, and donate some of the proceeds to support of a local ministry for the poor.

This year Mark and his parents attended this event as guests of their neighbors. One might think that this sort of event wouldn’t hold a lot of interest for a rambunctious young man like Mark, and one would be correct, but the organizers are aware of this so the couple the event with a barbecue and games.

Most of the crafts exhibited are woodworking and sewing; things which are not necessarily very delicate. One of the exhibitors by the name of Peter Paulson, however, does ceramics. He’s known for his porcelain pigs. He calls them his porcelain porkers.

At one point during the narrative, my friend paused and said, “Peter Paulson is pretty persnickety about his porcine pottery.”

Some people.

Paulson numbers his creations. There are little index numbers on each that are hidden in obscure spots. Each particular series is numbered consecutively from 1 to the highest number than he makes. He is a true artist in the sense that he continues making his art until the spirit leaves him. The pigs are each roughly the size of a baking potato, and he presents them partaking in various activities not usually associated with pigs such as riding bicycles, playing cards, and—somewhat disturbingly—eating bacon.

They are surprisingly delicate. They can be held in the palm of your hand, but you wouldn’t want to drop one even from a short distance. Paulson had a whole series on display at this particular event.

It happened on a Saturday afternoon. The men of the church had spent the morning smoking meat for the barbecue. It had been very sunny. The children had been playing outside and the women had been exploring the crafts. Then, as so often happens in the spring, it clouded up and began to rain. The children who had been enjoying themselves a great deal in the out-of-doors were forced inside by the rain.

They say that a butterfly flapping its wings in China can cause a hurricane in the Caribbean. We are told that for the want of a nail a horse was lost. That is all well understood. How much more well understood is the combination of rambunctious, over-stimulated boys, gymnasiums, and basketballs.

Someone at the church had left the basketballs out in a huge trash can on wheels. One of the boys—somehow ignoring the craft booths set up on the basketball court—took one of the balls and launched it from half court toward one of the goals.

The ball ricocheted off the rim and landed directly in Peter Paulson’s booth. The table fell with a thud, crushing every single porcelain porker on it. There was nothing larger than a postage stamp left of the porcelain pigs.

It was a mess.

As the pieces of porcelain pig were swept up, Peter Paulson, the pastor, and the holder of the insurance policy had a discussion. The insurance company would pay. The pigs were valued at $50 apiece. The question was: how many were there. The answer was: Peter didn’t know. He knew how many he’d sold, but he didn’t know how many he’d brought.

“It was an entire series,” he said. “It could’ve been 150 or it could’ve been 500. I just don’t know.”

“Well,” the pastor said, “I know how we will find out. We will make these boys put the pigs back together. The punishment will fit the crime.”

The people standing there looked at the bucket of porcelain crumbs and at the group of boys between the ages of eight and 14. When they looked back at the pastor, they had a look in their eyes, but it wasn’t belief.

Then a small voice with a country accent spoke up. It was Mark.

“Preacher,” he said. “We don’t have to do all that if you just need a general idea.”

“Son, I know you…” the preacher began, but he was interrupted by the mother of Mark’s neighbors.

“Listen to what he says,” she said gently.

“Okay,” the pastor said. “Go ahead.”

Mark then looked serious.

“If you just need a general idea,” Mark said, “I can tell, but I have to look through the pieces.”

“Well,” the pastor said, “you can do that while we are thinking.”

Mark and the rest of the boys went to the bucket containing the porcelain fragments. They each dug out a handful, pour them out on a table, and began looking through them. It took about twenty minutes, mainly because the boys would occasionally start tossing pottery fragments at each other.

At the end, Mark wrinkled up his nose in thought, and then stood up and walked over to the pastor.

“There were 160 give or take 10,” he said.

Peter Paulson got a hard look on his face.

“We just decided that to say there were 500,” he said. “So thank you for your trouble young man.”

Paulson was trying to take the attention away from Mark, but Mark wouldn’t let him.

“But that cain’t be right,” he said. “The error cain’t be that big.”

At this point, the insurance man was showing some interest.

“You know,” he said, “we probably could just pay an expert to weigh the fragments, but the cost of that would have to be deducted from your insurance settlement.”

Paulson then got a funny look on his face.

“You know, now that I think about it, there were about 160 pigs.”

My friend was grinning from ear to ear when he finished the story.

“What I’d like to know,” he said, “is how the boy knew how many there were just be looking at a few serial numbers.”

“I think that he used something that we call the German Tank Problem,” I said. “Back during World War II, the Allies wanted to know how many tanks the Germans were producing, so that used this formula.”

I wrote it on a napkin for him. It was: N=m(1+1/k)-1.

“The N is the estimated number of tanks, k is the total number of captured serial numbers, and m is the largest serial number. You can get a rough estimate of the area by dividing N by k. My guess is that Mark and his friends found fifteen serial numbers and the largest of the numbers was 150. Mark knew there couldn’t have been 500 because the margin of error isn’t that big.”

I was curious about something.

“So, how was Mark rewarded for this?” I asked.

“He was spanked,” my friend said without hesitation. “He was the boy that shot the basketball to begin with.”

Chasing the dollar

2011-04-01T11:00:00.000-07:00

I wrote this a number of years ago and it had never found its way to this blog.

Chasing the dollar

By Bobby Neal Winters

Before I begin this story, I have to make it absolutely clear that my Aunt Vidalia was always a good Baptist. By this I mean to say that she loved God, loved her neighbor, and took care of those in need.

On the other hand, she never let anything as abstract as the Church Covenant get in the way of her taking a couple of trips to Vegas per year and availing herself of the facilities while she was there.

Vegas and Aunt Vidalia were made for each other. In Vegas, you have a city built in a desert with grand edifices creating a beautiful illusion. With Aunt Vidalia, there was a tall hairdo, fake cherry-sucker red finger nails, and enough costume jewelry to have a yard sale.

The difference between Vegas and Aunt Vidalia is that if bulldozers were to push down Vegas there would only be desert underneath, while there is a very real person at the heart of Aunt Vidalia.

Vidalia’s husband died some years back and after a period of grieving, she took up with a man called “Bum.” This is a case where the name Bum is deceiving. Bum is short for bumpkin and is a name that was stuck on him when he first entered the army out of the deep Oklahoma backwoods. Bum took to the military like a duck takes to water and retired as a Lieutenant Colonel, and with the name Bum glued to him all through is career.

Bum retired early so that he could spend time with his wife and this turned out to be a good thing because after about ten years she passed away. He’d love her and this had left him lonely.

He met Aunt Vidalia and immediately recognized her as a person of worth in spite of the surface accoutrements that others might’ve taken as signs of shallowness. While she was in many ways the opposite of his late wife, who was reserved and understated in her choice of clothing, he discovered that Vidalia had the same inner strength in her that had enabled him to rise from an Okie in the backwoods to the rank of Lieutenant Colonel.

Little did he realize that this commonality they share would lead to a loss of domestic tranquility.

The occasion of this loss of tranquility was a party in the nicer part of Shawnee, Oklahoma. There would be those who’d deny that such a place exists, yet “nicer’ is a relative term and I will stand by it. Bum had been invited to this party by an old high school friend of his who’d gone into the oil business and made good of it.

It was one of those parties where liquor flowed rather freely. I don’t want to leave the impression that anyone was incapacitated in a noticeable way, but there came a stage of the party when many of those present were not in total possession of their better judgment.

It is at this point that a new character enters the picture. He was the son of the party’s host and I will refer to him as Jack. Jack was a graduate student who was majoring in math and was visiting his parents over the weekend.

My Aunt Vidalia is a friendly sort and struck up a conversation with him.

“What are you learning in school?” she asked. “Is it all just faster ways to multiply and divide?”

“I’m studying Game Theory,” he said.

By this time there were other people listening in on the conversation.

“Game theory,” Aunt Vidalia asked, ‘what’s that?”

“It’s just what it sounds like,” he said. “We study games.”

Aunt Vidalia was skeptical. She’d grown up poor and had to work all the time so she didn’t see much use in that.

“Studying games?” she asked. “What good is that?”

“Let me show you,” he said as he reached in his pocket and pulling out a dollar. “I can sell this dollar and make money on it. I’ll auction it off to the highest bidder. The only thing different from a regular auction is that both the highest bidder and the second highest bidder pay. Who will start the bidding at one penny?”

Aunt Vidalia likes games and the notion of getting a dollar for a penny was more than she could pass up.

“I’ll bid a penny for that,” she said.

Unfortunately for her, that sounded like a good idea to other folks at the party too. There were bids of 2 cents, 3 cents, and a nickel in quick succession.

Aunt Vidalia chimed in at 10 cents and again at a quarter.

Had she been paying closer attention, she would have noticed that people where dropping out of the bidding. She didn’t notice until she bid 50 cents that she and Bum were the only folks left bidding.

The slower pace of bidding gave her time to think. She would only be making 50 cents now if she got the dollar, but if she came in second she would lose 50 cents. If only Bum would drop out she would be all right.

“Fifty-one cents,” Bum said. He smiled and took another sip of the scotch and soda he’d been working on.

Clearly Bum was having a good time, but she wanted the dollar now. She looked him square in the eye.

“Ninety-nine cents,” she said and took a swallow of the Bloody Mary in her hand. She thought that she’d get her dollar, make a penny for her trouble, and Bum would get stuck holding the bag.

Now it was Bum’s turn to look her in the eye.

“One dollar,” he said.

If looks could kill, he would have dropped dead, but they can’t so he didn’t.

Bum and Vidalia were so busy looking at each other, they didn’t see the look on Jack’s face or anybody else’s. Jack was grinning like a possum, and everyone else was as well. If Aunt Vidalia had seen the way Jack was smiling, she might have smile herself and let laughter end the game.

As it was, she didn’t.

“One-dollar and fifty cents,” she said without batting an eye.

“Are you crazy?” Bum asked. “You are paying a dollar and fifty cents for a dollar?”

“No,” she said, “I’m paying 50 cents to stick you for a dollar.”

He paused just long enough to internalize that.

“Two dollars,” he said.

They fifty-cented back and forth until Bum bid $10.50. Then Aunt Vidalia remembered she only had 10 dollars in her purse.

“You can have the damn thing,” she said.

Bum gave Jack the ten-fifty, Aunt Vidalia gave him ten, and Jack gratefully accepted and handed the dollar to Bum. Bum offered it to Aunt Vidalia, and she told him where to go.

“I think it’s time to go home,” Bum said. So they did.

In silence.

It’s been about a week now, and they haven’t spoke again. I hope it all turns out all right.