Excerpts from the 2020 LeBow Preliminary Round

On February 3rd and 4th, sophomores and juniors participated in the preliminary round of the 2020 Robert H. Lebow Oratorical Competition. While the student body had the opportunity to watch the six finalists in the all-school assembly, they did not have the chance to hear all of the contestants perform their speeches and listen to their stories. All of the messages presented by the participants are important to the community, and as a result, excerpts from a few first-round speeches have been provided. 

Brian Li: “The Neglect of Humanities in the 21st Century”

The humanities serve as the foundation for human civilization. They teach us how to think critically and creatively. They facilitate empathy and compassion. We communicate more clearly, understand cultural values, and are introduced to new perspectives through the humanities. Humanistic education has been the core of liberal arts since the ancient Greeks, challenging students through art, literature, and politics. The humanities bring clarity to the future by reflecting on the past. Perhaps most importantly, they allow us to explore and understand what it means to be human. We need, now more than ever, to be able to understand the humanity of others. And while STEM is undoubtedly beneficial for humankind, we cannot sustain the neglect of the humanities for much longer without suffering severe consequences. With revolutionary technology like AI and genetic engineering, the humanities are essential to ensure that we progress ethically and morally together. 

Andrew Wong: “Where do you see yourself in 10 years?”

A question many of us in this room have heard in countless interviews, and yet it is a question that can reveal so much about ourselves. So, where will we be in 10 years? In 10 years, we will have found ourselves going through the entire college process and graduating from Pingry. We will go to and then graduate from college. We will have the rest of our lives splayed out in front of us, ready to grasp in our hands, as we enter the real world, ready to be the next generation of business people, engineers, doctors, lawyers, politicians, world leaders, and so much more. That answer seems pretty easy and straightforward, right? In reality, not so much.

Lauren Drzala: “An Uphill Battle”

As time went by, I was falling behind on work because I could not write at all, leaving me frustrated in school. In addition, my mental health was at an all-time low. I began to isolate myself, believing that no one could really understand how I was feeling. It was like I was falling and no one was there to catch me. On top of that, I was told I could not play my sport this winter, nor could I play the piano, an instrument that I have been playing since I was nine. I felt out of control and just had to watch the train wreck happen. I thought my friends had moved on so I tried to as well, but I just felt stuck. Left behind. I was in this mind set for a while, but it wasn’t until I realized that I could not give up on myself and settle for this empty feeling that my life started getting back on track. This triggered an uphill battle to try and climb my way out of the dark. 

Aneesh Karuppur: “Straying from the Tune”

We are always told that “small steps will lead you to your goal” and “you won’t even know how much effort it really takes if you do it one step at a time.” But people forget that for this advice to work, you have to actually be looking at your feet and making sure that every step is in the right direction. Otherwise, you simply aren’t going to notice until you’re too far away from your goal to make a correction.

Fractals Math Problem

Fractals Math Problem

Fractals are self-similar. The Sierpinski Triangle is an interesting fractal pattern. You can build one like this:

  • 1. Start with an “upright” equilateral triangle (as seen in stage zero)
  • 2. Place an upside-down triangle in the center of any upright triangle
  • 3. Repeat Step 2 infinitely 

Write a recursive formula that can model how many similar triangles you can count in the figure in a given stage. The figure above will be helpful in finding the pattern.

Extra Challenge: Turn that recursive formula into an explicit formula (i.e. find S(n), where n is the “stage” and S(n) is the number of triangles)

Click for Solutions!

S(0) = 1

S(n) = 3S(n-1) + 2

Base case is 1 because there is only 1 triangle in Stage 0.

Look at the jump from stage 0 to 1. When you add the white triangle, you essentially have 3 smaller versions of Stage 0, along with the central triangle and the entire triangle. That’s 3*1+1+1. 

Stage 1 to 2 follows the same pattern. 3(4) + 1 + 1.

Generalize the pattern to obtain your recursive step, S(n) = 3S(n-1) + 2.

In order to make an explicit formula, try expanding the recursive step. 

S(n) = 3S(n-1) + 2

Replace S(n-1)

S(n) = 3( 3S(n-2) + 2 ) + 2

Replace S(n-2)

S(n) = 3( 3( 3S(n-3) + 2) + 2 ) + 2

Simplify to make the pattern more clear.

S(n) = 3^3 * S(n-3) + 18 + 6  + 2

If you keep on replacing k times you find the pattern:

S(n) = 3^k * S(n-k) + 2 + 6 + 18 + …  2*3k-1

We can get rid of S(n-k) by using our base case of S(0). When k=n, S(n-k) is 0. Thus…

S(n) = 3^n * S(0) + 2 + 6 + 18 + …  2*3n-1

Final Answer:

S(n) =3^n + ( 2+6 + 18 + … 2*3^(n-1) )

Or more formally, using sigma notation:

Pingry Teachers Crossword

Pingry Teachers Crossword

Click for Solutions!

Across

2. Honohan   4. Jolly   8. Crowleydelman   10. Dunbar

Down

1. Grant   3. Horesta   5. Leone   6. Decatur   7. Jenkins   9. Trem

Epidemiology Math Problems

Epidemiology Math Problems

In these confusing times there’s a certain sense of power that comes with understanding, or at least trying to understand, the dynamics of disease spread and epidemiology. I certainly felt that as I’ve looked into the mathematics disease modeling over the course of this quarantine––a field that is absolutely fascinating, empowering, and daunting all at once. I recommend checking out Julian Lee’s article on his original disease modeling application, which describes the effects of visiting friends in a quarantine (his application is a randomized agent-based model as opposed to the more top-down mathematical models you’ll see here).

In this article, I present three epidemiology thinking problems, in ascending orders of difficulty, meant to put you in the position of the disease modeler. Have fun!

Fig: A graphical representation of an SIR (Susceptible-Infected-Recovered) Disease Model. The green curve is the curve that our social distancing efforts are “flattening.” This is the classic, top-down disease modeling approach

1) In epidemiology, R0 (“the basic reproductive number” or “r-naught”) is defined as the maximum number of new cases expected per infected person. The R0 of COVID-19 has been estimated to be 2.28 (Zhang, et al). An infected individual is expected to cause ß new cases per day. This is essentially the “birth rate.” Meanwhile, δ is the expected proportion of cases dying on a given day. This is essentially the “death rate.” Describe R0 in terms of ß and δ.

2) Does R0 tend to decrease, increase, or remain the same as a virus grows in an isolated population? Why or why not?

3) Consider the following differential equation to describe the spread of COVID-19, where ß and δ have the same values defined in problem 1. 

This is essentially a fancy mathematical way of saying that the rate at which the infection spreads is dependent on the number of people infected as well as the difference between the virus’ birth and death rates. In what ways does this model oversimplify real-life disease spread?

(Problems inspired by Rob J. de Boer’s Modeling Population Dynamics: a Graphical Approach, available here: http://theory.bio.uu.nl/rdb/books/mpd.pdf)

Click for Solutions!

1) R0 = ß/δ  (1/δ is the mean number of days someone with the virus is expected to live.)

2) R0 decreases; as an infection grows, there will be more infected individuals around a given infected person, meaning that there are fewer people to infect. 

3) There are a lot of answers to this question, which highlights the limitations of this top-down mathematical modeling. Here are a few. 

  • The ß value should not apply only to the infected population, but also to the healthy susceptible population. 
  • This model does not take into account the recovery rate.
  • The virus’ birth rate will change, not only depending on the number of people with COVID but also depending on how well the population responds (social distancing, hand-washing, etc)
  • The death rate will change based on what treatments are available.