Math's Rich Pageant
by David Francis
Any mathematician will tell you that there is an intrinsic beauty to mathematics. That, of course, is why they’re mathematicians. They have this vision that allows them to see things that most people cannot. Of course, normally, society prefers to provide professional attention and care to people who see things that no one else does. Its always been a mystery to me just why mathematicians escape this special attention.
Anyone reading this might suspect that I am just another person who struggled in every high school math class and developed a deep-rooted hatred for this so-called “Language of the Universe” along the way. If only that were true! How much easier my life would have been if I had just said goodbye to mathematics after high school, instead of actually earning a B.A. in this subject.
I suppose everyone has some sort of relationship in their past that they are not proud of. Mathematics is mine. Today I am ashamed of the way I let mathematics mistreat, abuse, and humiliate me. There was never any question that we were ill-matched. Math was always my worst subject. After some beginning Algebra, I managed no better than C's for the rest of my high school career. For some reason, I kept after it and pursued it as a major at college. I guess I was just hoping that some day it would change and start treating me right. But once I started college, there was only an occasionally inspirational B or B+ appearing in a endless sea of C's. I did manage to graduate, but since my marks were so mediocre, I wasn’t so sure that I would be able to make a career out of my studies.
As I was pondering my future in mathematics one day, I happened to be in the main lounge of the math building. Sitting in a chair nearby was a professor and a group of grad students. The students were sitting around the professor listening to some story he was telling. The professor reached the end of his story and with a huge grin and a laugh that he could barely contain, he exclaimed: "So I concluded that x squared plus y squared had to be equal to zero!" At which point everyone in the room burst out laughing and kept laughing for several minutes.
That’s when I finally realized that mathematics wasn’t for me. The fact is, I knew enough math to understand why a mathematician would think that was funny in the given context. But at the same time I realized that I would never find it funny. I finally realized that something was going on in the heads of true mathematicians that I, for better or worse, didn’t have. Still, even after this monumental turning point in my life, there are regrets. After floundering for four years in mathematics, I wish for moments when I can use the skills I developed as a mathematician. When these moments do pop up, I am always eager to perform. At a restaurant, at the end of meal, I always amaze my friends with my tip calculating ability. I am also very good at calculating sales prices when shopping. These moments of glory help me deal with my need for mathematical fulfillment, but I have always craved for something bigger. Something I could do to really impress people.
Two months ago, that opportunity arrived.
A past co-worker called me one day at work. After a few pleasantries, he explained that he had a bit of a mathematical dilemma and was hoping I could help him out. I, of course, was more than eager. I assumed he had a geometry question or some sort of thing. As it turned out, he had something a little bit bigger.
Recently, he had joined a committee that was overseeing election of the county fair queen in his area. The committee had to choose judges, decide events, qualify scoring rules, and basically oversee all the various pageant administration details. It was on the matter of scoring that he wanted my help.
The events were pretty much etched in stone in that county. There was to be an interview event, a talent event, an evening gown event, a speech presentation, and lastly, a penny vote.
With the first four, he had no problem. These were your basic pageant events and could just be scored from 1 to 10 with an equal weight for each event. But the fair’s unusual and long-held tradition of the penny vote was a bit more tricky, and it had brought some scandal to the fair in the past.
Here's how it worked. Containers were set up in a booth at the fair and labeled with each contestant's name. As people passed, they could vote on a contestant by dropping a penny in the container. The problem in the past had been that people would dump exorbitant sums of pennies into the container of their favorite contestant, and not coincidentally, the contestant with the wealthiest background usually managed to win the penny vote. This, of course, tended to produce some ugly sentiment around the pageant, but it also tended to produce a fair amount of revenue for the fair.
So my friend had a bit of a problem. With the money it brought in, the organizers of the fair did not want to completely scrap the penny vote, but, on the other hand, reform for this practice was long overdue. What he wanted to know from me was whether there was some way to tone down the effect of the penny vote while still keeping it as a factor. I suggested that he just make its scoring a smaller percentage in the overall judging. He said he had thought of that, but that there was still the problem of how to score the penny vote. For instance, he said, there had been times in the past when one contestant had earned 50 times more pennies than another contestant. How, he wanted to know, could they score this even on a scale of 1-10, without such a person completely walking away with the event?
I saw his point immediately. Following the example my friend gave, imagine that each contestant typically garners about 400 pennies, but that one girl has an extremely generous benefactor that deposits 20,000 pennies. This amount is so skewed from the average that it is hard to assign points between 1 and 10. If we give the 20,000 a score of 10, then the 400 pennies only earns a score of 0.2. You can imagine that the announcement of scores with one 10 and every other contestant earning less than 1 point would not go over too well. My friend assured me that there had been trouble in the past.
Fortunately, Probability & Statistics was a class in which I had actually earned a B+. Furthermore, I had also helped tutor an introductory probability course at one time, so I actually had some idea of how to proceed with this problem. As any mathematician could tell you, the problem here was not a seriously difficult one. All it really called for was a function of some kind that would reduce the growth rate of values and keep them more closely grouped together, and thus easier to assign point values. For instance, if we merely take the square root of the 20,000 and 400 pennies in the example above, we get 141 and 20 approximately, while still maintaining a fair relationship. This reduces the value gap from 19,600 pennies to 121 pennies. In this case 141 earns a 10 and the 20 earns a 1.4. This is an improvement, although still not enough to make scores come out pleasingly. But it gave me something to go on. I asked my friend if I could think it over for awhile. He replied that the committee didn't meet again for another week. I told him I would certainly have an answer by then.
I went to work on the problem that evening with sincere earnest. I was delighted at the chance of putting my math to some use, and to be doing so on a civic level thrilled me to no end. In fact, looking back, I guess I got a little carried away. What I produced was an entirely sound mathematical construction for producing acceptable results. I was really very proud of it. I won’t describe the equation here in total, but what it did essentially was first calculate the average of the total scores, take into consideration the highest value difference between all the points on either side of the mean, and then produce an average based on how the majority of the contestants scored. This number was given a score of 7. Using a modified logarithmic function, scores leveled off quickly on either side of this point. Thus, the person who earned 50 times more than anyone else would still earn a 10 and win the event, but the others would follow with scores more like 8.2 or 7.4 or 6.3, etc. rather than scores like 0.2 or worse. In other words, it produced a bell curve with the majority of scores clustering around the 7, and the number of extreme values tapering off quickly on either side.
If I had been a better mathematician, I probably could have made the process much simpler, but as it was, my scoring system looked pretty forbidding on paper. When I showed it to my friend, he was skeptical. But after showing him how well it handled some examples and even some event results from the past, he was won over. Still, he asked that I come along to the contest to make sure everything went smoothly. As far as I was concerned, this was my crowning achievement in the world of mathematics, and I wasn't going to miss it for anything.
I was excited that evening to tell my wife about our weekend plans. She was less inspired than I had expected.
"You want me to waste a Friday evening at a 4-H fair pageant?" she asked.
I was somewhat put off by her attitude. To me it seemed she had missed the real adventure here.
"Well, yes, it appears that the most glorious moment of my life just happens to be at a 4-H fair pageant. If it would be stooping to include yourself in this momentous event, then I will find a woman who appreciates a good mathematician and take her instead!"
This kind of threat, of course, was too much for her to bear, and with only a brief rolling of her eyes, she agreed to go. I felt bad, of course, having been so hard on her, so I quickly added:
"And don’t you worry: you’re my 4-H queen!"
The night of the pageant was a crisp autumn evening, and it was really a very pleasant time to be on the county fairgrounds. The pageant was only one event going on at the fair, and we spent most of the evening strolling around. I really didn’t want to spend a lot of time at the pageant. Rebecca had fairly noted that pageants aren’t much fun, and I had always found them somewhat disturbing myself. In fact, for some reason, it is especially disturbing at a county fair where you can stop at one booth and see high school girls walking on a make-shift stage in evening wear, and then walk over to the next booth and see cows, pigs, and sheep doing much the same (without the evening wear) for auctioneers and farmers.
But I wasn’t bothering myself with these thoughts on that night. The fact was, the most important part of the pageant for me was going to be the scoring. So while the rest of the contest was going on, I was keeping one eye on the penny vote booth. There had been some interesting occurrences there. Each contestant had been given a silver painted milk can about 4 ½ feet tall. On at least three occasions I saw someone walk up and pour whole bags of pennies into a container. Then, just as the pageant was winding up, I saw a middle-aged man with two high school aged boys head towards the penny booth. Between the three of them they were carrying what looked to be a very heavy feed sack. They were panting heavily from the weight but laughing as they reached the booth. They put their shoulders into it, and dumped the bag, filled with pennies, into a milk can situated near the middle of the pack. The milk can was so full after this that a little peak of pennies had risen up out of the lid. The three men admired their work, and laughing again, gave each other high fives and then melted back into the crowd.
I had a little chuckle myself. No doubt they thought they had busted up the game with their little shenanigans, but they had not reckoned with my mathematical prowess. I quickly pulled out my formula and began to explain to Rebecca how easily my theoretical construct would dismiss this problem. "Ha!" I said. "They don’t realize that the more pennies they pile on, the less each penny is worth! Here, I bet we can get a rough estimate of the logarithmic curve right now…"
Rebecca patted my hand with a polite smile and then gave a furtive glimpse at her watch.
A voice over the loudspeakers announced that the pageant’s events had been completed and that in a few minutes they would be announcing the winners. Three large men in flannel shirts and caps came up to the penny booth with dollies and carted the penny-laden milk cans off to be counted. Actually, they didn’t count them—they weighed them. Counting the pennies would have taken hours. Years ago they had learned to weigh the pennies and then divide the weight by the average weight of a penny. I thought that was pretty clever, and I realized I wasn’t the first mathematician performing his civic duty at this rich traditional autumnal festival.
In less than half an hour, the announcement came over the loudspeaker that the results were in. We moved over with the rest of the crowd to the pageant stage where the contestants were already on the stage, beaming with hopeful happiness. On the other side of the crowd semi-circled about the stage, I saw the middle-aged man and his two boys push their way to the front. They still had huge grins on and could hardly keep still while waiting for the results.
The announcer began to pronounce the results from the various events for each contestant. I didn’t really pay much attention, but just clapped along with every one else. Pretty soon, most of the girls had stepped into the background and only three remained at center stage. As it happened, these final three contestants had made a real race of it and with an admirable sense of high drama, the announcer declared to the audience that he had never seen such close balloting before. Most of the audience ooohed with appreciation, but the three guys on the other side of the crowd immediately started slapping high fives again. They knew that if the score was going to be close, their work at the penny booth would deliver them the contest.
The scores were close, and, much to the surprise of the three high-fiving men, the penny vote was pretty close too. Oh, they won it easily enough. As I recall they set a new record with 27,000 pennies for their contestant. The second highest amount of pennies was only 1,000, and that girl hadn’t even made the top three anyway. When the 27,000 pennies were announced along with their 10, the three men just about did a jig where they stood. Then the final girl was announced the overall winner because her 327 pennies had earned her a 7.9 and that was just enough to edge out the top penny winning girl.
I have to say I enjoyed watching those guys faces turn from petulant delight to dumb disbelief. Yes, it was fun for a second or two, but as their faces went ahead and kept turning to new forms like displeasure, and then anger, and then rage, and then moving right along to violent insanity, it became harder and harder to get a good laugh out of the situation.
There is an old story about mathematicians that is popular in academic circles. It goes like this. A an engineer, a chemist, and a mathematician are all staying at a hotel for a conference. The hotel is pretty cheap so there are no fire extinguishers in the rooms, only water buckets. In the middle of the night the hotel has a power surge and every TV in the place catches fire. The engineer wakes up, sees the fire, and being an engineer, his only thought is to just solve the problem with whatever means available. He quickly grabs the fire bucket, runs to the bathroom sink to fill it up, and then starts wildly splashing water on the TV until the fire is out. In the next room the chemist wakes up to see the burning TV, but with a mind for exact quantities and specific measures, he first sits down and calculates the size and heat of the fire and then runs to the bathroom filling the bucket with only just enough water to douse the fire. In the last room, the mathematician wakes to see his TV on fire. He sits up, looks at the TV, sees the bucket, and sees the spigot in the bathroom. Then, satisfied that there is a solution, he goes back to sleep.
Back at the fair, I was in a similar situation. I had failed to realize that no matter how precise and mathematically beautiful my penny calculating formula was, it wasn’t going to please everyone. As the new county queen received her crown, the three men weren’t calming down. As the festivities ended and the crowd started to disperse, they were still standing in place with dark faces. Finally, the middle-aged man, who I later discovered was the runner-up’s father, waived his two boys along and they stomped over to the judges’ table. I watched them engage in an animated conversation with the respectable magistrates. Pretty soon, the friend that had first enlisted my services was called over. The conversation continued for awhile and then finally over the loudspeakers I heard a voice say:
"Would David Francis please report to the judges’ table."
Well, anyone who knows me will realize that I had had my fill of civic duty at this point. I quickly prepared to vanish into the throng, but as I turned to drag Rebecca with me (a big mistake—I should have just left her, but even when my civic duty is at low ebb, my sense of chivalry lives on), my friend spied me and started to waive me over to the table.
"It might be a good idea," I mentioned to Rebecca, who hadn’t taken much notice of any of these affairs, "to go on to the car now. And you might want to have the motor running…"
And with these last brave words, I moved over to the judges’ table.
Among the three men, the father had taken over at being the absolute angriest. His two sons looked pretty irate, but they were remaining respectfully quiet and letting the patriarch do the necessary ranting and raving. As I approached I heard something like:
"I don’t give a damn about your formula! I spent $250 on pennies and someone else spent three and won this thing! Now I know that 27,000 is at least ten times more than 300, so somebody tell me how someone else can get an 8 with a tenth of what we got?"
The man was yelling at my friend and when he saw me approaching I could see some relief appear on his face. He quickly interrupted the man’s diatribe and pointed to me.
"This is the man who helped us with the penny vote this year." he said soothingly as he guided me into the man’s swath. "I’m sure he will be able to explain the results to you."
"Well, I’ll just bet he can’t!" the man responded. And just as he finished, two more groups that he been attracted by the yelling stopped by. They also wanted to know why the scores were so skewed, and they didn’t look much happier.
"Well, its really very simple," I said in my finest pipsqueaky voice that always descends upon me when I need it least. "The scores were converted to a statistical spread to keep the voting fair. You see, the real power of it is the logarithmic function…"
I handed my copy of the formula over to the group of protestors to prove my point. The father scanned it over quickly. The symbols he saw there only seemed to make him angrier. He quickly countered my argument with one of his own.
"This is bullshit." he stated.
"Well, its all entirely mathematically sound." I replied, realizing as I did that even I might want to hit someone who said that. Sure enough the man flew into another rage and this time he was backed by the newcomers. My friend was turning deathly white and the honest judges were obviously preparing to turn the law of the pageant over to this angry mob. It was at that moment I decided to abandon all of my fantasies about mathematical prestige, and instead return to what I did well.
The fact is, I have always been a first-rate liar. I’m good at it and I enjoy it. That’s what really sets me apart from the run of the mill liars in this world. Most people don’t like to lie. It makes them uneasy. Not me. I love it. Being caught in a lie makes me uneasy, but if I can pull one off, I will never give it a moment’s worry. As far as lying to save my skin—well, in those cases I can be in rare form.
"Would it interest you to know," I said when the man stopped to catch his breath, "That they use this same formula for the Miss America pageant?"
This stopped the mob for a moment. The father eyed me doubtfully.
"Like hell," he said. "They don’t even have a penny vote for Miss America."
You might think that would have stumped me. Oh no. When I start a lie you can bet I will finish it. The Muse of Deceits just descends upon me and I spontaneously create untruths.
"Oh no, not a penny vote." I said. "But they have the swimsuit competition. In recent years the feminist movement has put pressure on the pageant to get rid of the swimsuit part. Well, they have always resisted that, of course, but they have had to make some concessions."
Do you see the beauty of this lie? I’m not bragging. I don’t take credit for it. These things just come to me when need arises. But you see, with just the mention of the feminist movement, I had already started to deflect some of the blame from me to a more general, faceless enemy. I continued.
"Yes, the swimsuit competition doesn’t receive the same voting system that the other events do. They use this very formula to damp it down. What? You thought I came up with this? Me? Oh no, I just took it from their official rulebook."
The father muttered to himself. He was still plenty angry, but he wasn’t quite as sure of himself. He started to wonder if he was dealing with something bigger than himself and that maybe a bunch of screaming at the judges’ table wouldn’t be sufficient to solve it. I thought for a moment maybe he would just throw up his hands and go home and write a letter to his congressman about this, but he wasn’t quite done yet.
"The Miss America Pageant doesn’t even have a rulebook!" he shouted as though if he said it loud enough it would become a fact.
"Doesn’t have a rulebook?" I replied, becoming calmer as I let the deceit flow through me, flow around me, and envelop me. "Then what is that thing sitting on the dashboard of my car, I’d like to know, and where did this come from?" I added pointing my formula. "No, okay, fine. Fine!" I continued, waiving my hands in frustration showing my genuine shock that he could doubt this. "Just wait here a second. There’s no point in arguing about this when I can prove it to you. I’m going to go get the rulebook from my car and I will show you!"
I stalked off with real anger. You see, the secret is to immerse yourself in the lie. Even I believed I was going to walk to my car, grab the fictional rulebook, and shove it in this man’s face. Of course, once I was out of their site, my senses returned. I broke into a sprint and headed for the car. My ever-trusty wife had the care revved and waiting. I jumped in.
"Step on it." I said.
"I knew something like this would happen." she said, pulling away. "How did you get away?"
"Never mind about that right now," I replied. "Can’t we go any faster?"
"Not without killing pedestrians."
"Well, let's keep our options open." I said. "Just so you realize it may be us or them."
I was rattled, but Rebecca coolly navigated us out of the fairgrounds and back on to the main road. I gazed back fearfully at any headlights that appeared behind us on the half hour ride home, but we made it back without incident.
It was a trying experience, but not without its rewards. I mean, while it certainly wasn’t received warmly, my formula actually performed very well. I can take pride in that, and now that I realize what trouble mathematics can lead to, I don’t resent the fact that I am second-rate math guy anymore. As long as I can maintain high caliber lying skills, I don’t need to bother with mere mathematics.