Monday, April 4, 2011

The porcelain pig problem

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.”


Julia said...

I've been loving this series! Let me guess -- you're teaching stats this semester?

Bobby Winters said...

Yes. I love stats. It lends itself to stories like this.

Hope you are doing well!