Monday, March 21, 2011

Mark N. Release

Mark N. Release

By Bobby N Winters

I am known among my friends to be of a mathematical bent, and they like to share the mathematical experiences they have with me. Most often this is in the form of forwarded mathematical e-mails with videos or puzzles or puzzles within the videos. Some of the time, however, it is of a more personal nature. Such an instance came recently when a friend of mine brought a story to me over coffee.

“You mathematicians,” he said, “can just turn up anywhere. There is no rhyme nor reason.”

“I am not going to disagree with you,” I replied, but, being cautious I added, “but what exactly do you mean?”

“Geography, socio-economic factors,” he said, “it doesn't matter you just turn up. That mathematical mind just pops up anywhere.”

“That's right,” I nodded. “The gift is just imposed on some of us capriciously.”

I didn't like the way he curled up one of his nostrils when I said gift, but I let it slide. I was curious as to why he'd started on this line so I thought I would encourage him.

“Why are you bringing this up at this particular time?”

“Well,” he drawled out, “it is because of one of my cousin's kids back home.”

My friend is from Arkansas. As a native Oklahoman, I like Arkies. We Okies like them in general because they are a group we can consider to be less sophisticated than ourselves; they reciprocate this. I like them in particular since we seem to share many of the same experiences and attitudes. He began a tale about his cousin.

This cousin was from the Release branch of his family who lived in the Ozarks of North Central Arkansas. If you've not been there, you may need to be told that the region is beautiful. There are many “hills and hollers” that simply invite you to drive through and enjoy their natural beauty. Many parts are so isolated and untouched that they might remind you of the old Wild Kingdom television series. You expect to run into Marlin Perkins and his assistant Jim on an expedition of some sort.

My friend's cousin has a family, and it is his twelve-year-old son Marcus Nathaniel who has been gifted with the mathematical gene, as it were. The family lives on the old farmstead where they have a few cattle, a few hogs, and a few more dogs than absolutely warranted. They get their water from a well; they do have electricity, but don't have natural gas. None of this is necessary to understand the mathematics that follows, but it does give you an ideal of the caprice of the mathematics gods, as it were.

The Releases don't have TV. Being rural, there is no cable and they have not yet opted for a satellite dish. They entertain themselves in more traditional way. Young Mark is fond of hunting and fishing as young men have done in that region for many generations, but he's added one more element to it: paint-ball.

Paint-ball, as odd as it may seem, marks a point of entrance for us to observe a bit of our young man's mathematical behavior. The occasion of this particular bit of mathematics was the arrival of some new neighbors.

This is a part of the world where not many people come and go. Actually, these days there are more going than coming, so it was a particular interest when a new family moved in next door. Here I used the phrase “next door” which I now realize might be a bit misleading. Next door in that part of the world can mean that you are only separated by a fence, but it can also mean you are separated by a fence and a wooded area and a creek and distance of about a quarter of a mile. In this case, it was the latter.

The family moved in and Mark's mom, as is the custom in that area, took them over a pie. When she went over she determined that the family was blessed with a large number of children. They were so active that she couldn't tell how many that there were.

Mark was overjoyed when he heard the news because there would be new kids to play with. He was less thrilled to learn that the the parents were fairly strict and didn't let the kids play with strangers and it might be a while before the Releases didn't qualify as strangers.

This annoyed Mark and helped set up the incident that follows which began, really, when he overheard his parents talking.

“It's a shame them being so stuck-up,” his father said. “It'd be good for Mark to have some more kids to play with.”

“Well,” his mother said, “they are really religious and there very careful with their kids. I can't blame them none. These days you just don't know. And it's not like their kids are hard up for playmates. They've got more than I can count.”

Overhearing this, Mark took it as a challenge. He loaded up his paint-ball gun and hiked through the woods, forded the creek and hid himself in the underbrush. And he waited.

Soon a group of children emerged in the back of the house within his sight and began to play. He waited until they were all occupied in their game before he took aim and began squeezing off rounds. Before his was done, he'd stained six of the children neon yellow. He then beat a hasty retreat back into the woods and across the creek, losing the larger of the children who were following him.

He put his paint-ball equipment away and waited until the next day, whereupon he repeated this process with blue paint-balls. At that time, he stained five of the children, two of whom he'd marked before. He could tell because the paint is really hard to get out and some of them were green afterwards instead of blue.

This time when he got home, there was a car in the front yard and a strange woman with an angry sort of look on her face in the kitchen. It was not nearly so angry a look as that on his mother's face. She looked directly at the paint-ball gun in his hand.

“All right, Mister,” his mother said, “what in the world have you been up to?”

He looked at them both and spoke matter-of-factly.

“They have about fifteen kids, give or take about 4.”

“What?” his mother asked confusedly.

“They have about fifteen kids give or take,” he repeated. “I caint tell you exactly because they move around so much and they look so much alike being kin and all, but there's about fifteen of them.”

The strange woman's visage changed from stern to amused.

“We've been blessed with twelve children,” she said with a smile. “Is that was this was all about?”

“Yep,” Mark said crisply. “Even if you won't let me play with 'em, I can still count 'em.”

My friend end the narrative with a smile on his face.

“After that,” he said, “little Mark got to play with the neighbor kids next door whenever he wanted to. I don't know if the mom decided that Mark was okay or it was a case of keep your friends close and keep your enemies closer, but they've been getting along well. What I don't know is how the kid estimated the number.”

“Well,” I said, sipping my coffee, “I can't be for sure, but I think he may have been using the Mark-and-Recapture method. This is a way field biologists estimate numbers. You may have seen nature shows where they capture animals and put tags on their ears. It makes for great TV. What you don't see is they come back later and capture some more. If they recapture animals which have been tagged they can then estimate how big the population is. If you want to know the formula...”

My friend stopped me before I could continue.

“I am sure that is fascinating,” he said, “but I've got somewhere I need to be.”


“I don't know,” he said, “but there's got to be somewhere.”

Since he didn't want to know, I will simply share it with you. If M is the number captured and marked the first time and C is the number captured the second and if R is the number captured the second time that had been marked the first time, then the estimated size of the population is N=MC/R.

As Mark had painted six the first time, five the second, and two of them twice this works out to fifteen. There is also a more complicated formula for the standard deviation and Mark figure that the true number would be within on standard deviation of the mean.

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