Trenched

The world passed by silently, the trees swaying softly in the light, humid breeze. Everything was green, brown, and gray with dots of color from the trash that littered the picturesque mountainside, completely different from the white sands and turquoise waters of Bali where we had just ventured from. No sky shone through the clouds here, even though it was 96 degrees out. No matter what the temperature, Java was always gray and muggy, and full of either annoyingly warm mists or shockingly cold downpours. Yet no matter what the weather, Java was always unmistakably beautiful.

I watched everything pass as the train moved sluggishly up the hill, glad that I had rid myself of my slugabed ways and gotten up early for this excursion.

“What about the cake?” Natalie, who had a reputation for logorrhea to maintain, yelled across the train, her voice an annoying white noise to me as I looked below, my forehead pressed up against the cool window pane, my coconut smelling breath condensed onto it as I breathed, the rain blurred my view slightly as it dripped sluggishly down the window. The same rain that helps the beautiful red flowers to ameliorate from the harsh brown earth. I could see the slums made of tin and cardboard with people around them trying their hardest to keep their houses up while the rain slowly broke down the walls of their homes. Next door the kids played in the mud, throwing handfuls of it at each other, enjoying the feeling of it in their hands and laughing as the mud caromed off each other. Suddenly the rural village passed out of my mind as I felt the world stop, my eyes replaying the most disturbing thing I had seen in my short life over and over again.
There, in the middle of the hill, was a girl lying alone in a trench of sewage, dead. Her hair was in knots around her body, stuck to the excrement which was starting to become watery in the rain. Her arms and legs were twisted out around her in unnatural positions that my inflexible body would never be able to be in. Her face was cut up, but the blood had long since dried into a rusty smear. What was only a split second of passing felt like an hour as my mind replayed the scene over and over again like a CD stuck on repeat. The image of her face, her arms, and her hair were forever imprinted into the walls of my mind, a constant reminder of how different my life was from the people's here, and not wanting to think about it, I attempted to discard it from my memory.

The petulant feelings I had felt in the past suddenly seemed miniscule in comparison to the horrors I had just witnessed. The fact that a girl, who looked my age, maybe younger, was lying dead in a trench was the most extreme wake-up call I had ever gotten. My life's hardships were like a grain of sand compared to the wave of issues the people here dealt with every single day.

When I first saw the girl, I had tried to shake the picture of her out of my mind, ignoring the lesson she had to teach me. But now, as the scene began to die out and we moved farther up into the mountain, I hugged the memory of her closer. Within those few seconds, passing like hours, I had experienced the most terrifying and clarifying thing in my life.

Monto Gordal Beach, Portugal

It begins with the sound of water, trickling softly, tumbling down into the pool. Suddenly it is swallowed up by something bigger, pulling it back as it returns to it's hole. It is climbing up in that way where you pull up a lot, then fall back a little, pull up, fall back, slowly making pace. Slowly reaching it's destination. As it gets to the top it begins to slow down. Somehow, in its slow state, it gains momentum. With a surge of energy pulls itself far up, almost to it's goal. New sounds emanate from the being as it comes up farther and farther.
"Plink, plunk, plop" it yells out in frustration as it falls back away, still not quite reaching it's goal, "Plink, plop, plop, plunk!" It gets louder and louder. You can hear the anger in its voice as it struggles to make it to its goal. Harder and harder it works, and still it can not make it. The low grumble turns into a growl as its anger shows. Louder and louder it screams. The rage of mother nature burns as it still almost touches the point. Will it ever make it to its destination?
One more surge of energy. One more breath. One more plea and it is there. It is over.
To hear the sound click here.

The Locker Problem

In the locker problem we had to find out which lockers, out of a thousand, were opened and closed after a thousand students changed the state of the lockers. The first person would open all the lockers, the second change the state of every second one, the third change the state of every third one, and so on and so on. After making a chart of about 50 lockers and going through the pattern, marking it each time it was open or closed as well as how many times its state was changed, my group found that every perfect square locker number was open while the rest of the lockers were closed. We also found that the lockers had a pattern, one open, two shut, one open, four shut, etc.

Looking at the data we found that every closed locker had an even amount of factors, meaning its state was changed an even amount of times. The lockers that were open in the end were the only ones with an odd amount of factors, and its state was changed an odd amount of times. This meant that only perfect squares had an odd number of factors, or an odd number of times their state was changed in the case of the problem. For example four has the factors two, four, and one, because 1x4=4, and 2x2=4, and you only count two once when you list the factors. This relates to the lockers because each locker number will be opened or closed by each of its factors once. Student one, two, and four are the students that will open or close locker number four because they are the factors in this problem. Four has three factors, so it was opened, then closed, then opened. In the case of another number, like 21, there will always be an even amount of factors, one, seven, three, and twenty-one, making it have four factors., i.e. being opened, closed, opened, and then closed by the twenty-first person, and never hit again. Only lockers hit an odd amount of times were open because, as seen in the diagram above (click on it to enlarge), when a locker was hit an even amount of times, it ended up being closed.

In the end of the problem we found out that there are only 31 perfect squares within a thousand giving you 31 lockers open, and 969 closed. In terms of finding a general rule in order to solve this problem further, it would be extremely difficult, even with the pattern found above (1 open, 2 shut, 1 open, 4 shut). The general rule is hard to find because as you extend the problem, the number of closed lockers increases, along with the number of perfect squares. So, if we had to solve this equation for two thousand lockers I would stick to using either the pattern, or just find the perfect square locker numbers within the total amount.