Sunday, October 26, 2008

Counting dots

In a small town like where I live, you get to know everyone. One of the people I’ve gotten to know is the local homicide detective Suzanna Doughcoup. Sue, as she is universally known, is only a part-time homicide detective as there are so few homicides in these parts to detect. In order to pull her weight, she spends a lot of her time finding lost dogs and running skateboarders away from the library steps.
I was headed into the library and met her just as she was doing the latter.

“You kids scram,” she told them, and then she turned to me. “Hello, do you have a minute?”

“Hello, Sue,” I replied, “is there something I can do for you?” I am always eager to help our local law enforcement folks as I never know when I might need a parking ticket fixed.

“I’ve got a potential homicide with no body and no budget to do much investigation.”
This is a sleepy little town, so hearing that got my complete attention as she explained the situation. It seems that a huge red stain was discovered on morning on the tiles in front of city hall. The stain was blood red which caused much alarm among the municipal workers as they came to their jobs that morning.

Since it was in front of city hall, the police arrived on the scene quickly and taped off the putative pool of blood with police tape. Officer Harley Jones was in charge of this part of the operation, and he is very fastidious. He carefully applied the tape so as to go from corner to corner on the tile. The shape of the stain was so regular that it fit within those straight lines almost exactly, as shown below.

When Harley arrived, the stain was still wet enough that he could measure its depth to be about one-eighth of an inch.

“I know you teach math at the university,” she said. “Is there anyway you could measure the volume of this to give us an idea of how much blood this would be?”

I said I would try. I went back to my office to work it out.

In general, calculating the volume of a liquid spilled uniformly over an area is not difficult. All one has to do is multiply the area of the spilled liquid by its height. When the area is irregular, however, that adds difficulty to the problem. In this case, since the liquid was spilled over a tiled area, it would be easy to get a lower bound for the estimate by counting the number of tiles in the stain and multiplying by the area of a single tile. This was made even simpler by the fact each of the tiles was exactly one foot by one foot.

As an initial estimate, I counted the tiles that were completely covered by the stain and found there to be 19 of them. However, in the act of counting, I notice that a large number of the tiles stained with blood were only partially stained, my under-estimate might be serious. I began to wonder whether the fact the stain had such straight sides might make the area easier to estimate when I remembered Pick’s Theorem.

Pick’s Theorem is due to German mathematician Georg Pick and has to do with calculating the area of polygons in the Cartesian plane whose corners have integer coordinates. The formula is simple. The area of the polygon can be calculated as follows. Count the number of points with integer coordinates that fall completely inside the polygon, add to this half the number that call exactly on the boundary, and then subtract one. This can be represented as

A=I+(1/2) B-1,

where A is area, I is the number of points with integer coordinates inside, and B is the number of points with integer coordinates on the boundary.

Having remembered this, I counted the number of corner points of the tiles that were completely inside the stain and found there to be 29. I counted the points that were on the police tape and there were 9. Applying Pick’s Theorem to this gave me a total area of 32.5 feet. A depth of one-eighth of an inch made the volume of the unknown liquid work out to be 0.34 of a cubic foot or 588 cubic inches. There are 231 cubic inches in a gallon, so this works out to be two-and-a-half gallons.

Since the human body has about a gallon and a half of blood, on the average, this was alarming. I phoned Sue right away and informed her of my findings.
“It might be a double homicide,” I said.

“I don’t think so,” Sue said. “I took closer look at the so-called blood. It turns out Harley got a little too excited. At that time of the morning, red paint passes for blood. The guys at the Sigma Alpha Pi Frat house got a little inebriated and tried to paint their Greek letters in front of city hall. They used up all their paint on the Sigma, but got it a little blotchy in the process.”
I looked and the red blotch and squinted.

“I suppose that accounts for the straight sides,” I said.

“I suppose so,” Sue replied. She sounded a little disappointed. “I went down to the Frat house and found two empty paint cans and a half empty one. Your estimate pretty well nails down their convictions. Somebody’s dad is going to have to lay out some cash for this one.”

“Good detective work,” I said.

“They can’t all be murders, I suppose,” she replied.

“Better luck next time.”

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