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Why I Love Math [Applications Collection]

Now that I'm well-past completing my various applications, I thought I'd share some of the essays I wrote on this blog. I know they're shorter than my usual posts, but since they're actually, you know, proofread and edited, they're probably better reading. I'll be putting various essays and other application materials up in lieu of a blog post proper for the next few weeks. Enjoy this essay about my appreciation for math.


Imagine you’re standing on a point. Place another point in front of you. Then rotate your body by 0.618... (phi, the golden ratio, minus one) of the way around. Place another point. Repeat. You’ve made something called the phyllotaxis spiral, which is also how seeds are arranged on a sunflower. Learning how seeds are arranged on a sunflower may not necessarily be useful later in life, or even be an intriguing fact to know. But to me, it is inherently beautiful: the interwoven spirals are in perfect harmony, like ropes woven together to form an object even water cannot penetrate.

A coloring-page version of the phyllotaxis spiral. I highly highly HIGHLY recommend you print this out and color all the even numbers, or all the multiples of 3, or something like that. It's beautiful. Source: https://www.mathrecreation.com/2012/04/phyllotaxis-multiplication-colouring.html

I have always been a math person. My parents tell me that as a toddler, I would skip-count by arbitrary numbers for entertainment. That isn’t why I like math, though they’re certainly connected: people showed me why math was cool because I was tagged as a “math person.” In my spare time, I read math books, go to math talks, and participate in math competitions. I love math because it is beautiful and elegant; because it is a way to quantify the world; and because there are many different interesting and creative veins of math.


Math is a way to quantify the world. It has applications all over science, architecture, and multitudes of other fields. One such field is music. I play piano, and while I am willing to admit that I don’t practice as much as I possibly should, it still brings me enjoyment and fulfillment. One area that I am more advanced in is music theory, because it is so strongly tethered to math. Chord structures are rooted in ratios; transposing a song is akin to multiplying a mathematical expression by a constant. Beyond simple analogies, the pattern-finding skills carry over, bringing with them the elegance I mentioned in the last paragraph.

Too often, society makes math about computation, when it should be about exploration.

Additionally, math can be an outlet for creativity. You may be familiar with the Four Fours problem, a math problem wherein you must use four fours to make as many numbers as possible. In fourth grade, I discovered a general solution to this problem, involving multifactorials, that will produce any integer.* I remember pacing around the blacktop and field, trying to find a method to produce more numbers—I had figured out many of them already. Suddenly, I realized the answer. At first, I was skeptical. Could that solution actually work? Then my skepticism turned to excitement. When I got home that day, I wrote up my findings. To date, I’ve never seen this method anywhere else.


I believe that math can be more than memorizing formulas and tedious calculations. For me, math is more than a class in school. Math is not about absolutes, about the right answer and the wrong answer. Too often, society makes math about computation, when it should be about exploration. Done right, math is about the satisfaction of discovery, inspired, but not sculpted, and the pride of sharing it with others.

For reference, here's the phyllotaxis spiral with each multiple of 5 colored in. Source: https://www.mathrecreation.com/2012/04/phyllotaxis-multiplication-colouring.html

–beautifulthorns



*I don't think I've ever posted the actual solution on this blog, so here it is. It's a bit jargony, but it should be accessible to anyone who understands how variables work.


The Four Fours Problem

The Four Fours Problem asks, with four numeral 4s and no other numbers, but any number of operations (addition, subtraction, multiplication, etc.), what numbers can we make? For instance, 3 is (4+4+4)/4. However, my solution only requires two 4s. Or, really, two of any integer greater than or equal to 3. We'll call that number k. But keep in mind that for the actual Four Fours Problem, k=4.


(By the way, we do need to use all four fours, but it's trivially easy to get rid of them in pairs, by simply adding k and then subtracting k. we can also get rid of one k in some more convoluted ways–for instance, or by replacing a k in the expression with √k*√k.)


How Multifactorials Work

You're probably all familiar with factorials–to take the factorial of a number, take 1 times 2 times 3 and so on until you get to that number. For instance, 6 factorial, denoted 6!, is equal to 6*5*4*3*2*1=720. But less famous is the double factorial, where we only multiply every other number. So 6 double factorial, denoted 6!!, is 6*4*2=48.


We can extend this idea to a triple factorial (6!!!=6*3=18) or a quadruple factorial (6!!!!=6*2=12) and so on. The general term for this operation is a multifactorial.


There's an important notation I want to introduce for large multifactorials. Say we want to take the decuple (10th) factorial of 18. We could write this as 32!!!!!!!!!!, but this is cumbersome. Instead, we write 18!_(10). However, for the Four Fours Problem, it's important to remember that we could write it longhand with ten exclamation points. So if I write18!_10, I really mean 18!!!!!!!!!!.


The Multifactorial Trick

Importantly, these large multifactorials start looking a lot like multiplication. For example, when we take 18!_10, we are only multiplying every tenth number, of which there are two positive options–18 and 8. So 18!_10 is just another way of writing 18*8. This multifactorial trick (or, some might say, cheat) is the backbone of our solution.


We might ask, when does this work? a!_(a-b)=a*b is true if and only if a is less than or equal to twice b. So 18!_(18-8)=18!_10 does produce 18*8, but 18!_(18-10)=18!_8 doesn't make 18*10, it makes 18*10*2.


The General Solution

We will now officially solve the Four Fours Problem, or, rather, the Two or More ks Problem, via the multifactorial trick. First, we define the variables.


The Variables

We let k be the number we are using to form the numbers. In the Four Fours Problem proper, k is 4.


We let x be the target number we are trying to form. We will prove that it works for all integer values of x.


We let y be any number that we can make with a single k that is greater than equal to 2x, so the multifactorial trick works. Just imagine factorialing k a bunch of times. In fact, ((4!)!) is already huge. But the point is, it can be as big as we want it to be. Note also that this is why the solution doesn't work for k less than or equal to 2. No matter how many times you factorial 2, it stays 2.


The Method

Once we have a value of y, we use the multifactorial trick to multiply it by x. We do this by taking the y-xth multifactorial. Then, we divide by y to obtain x. In other terms:


x=(y!_(y-x))/y.


In Action

For k=4 and x=11, we let y=4!=24. Then (y!_(y-x)/y=

(24!_(24-11))/24=((4!)!_13))/4!=((4!)!!!!!!!!!!!!!))/4!=11.


The Wikipedia page on the Four Fours Problem lists 113 as an impossible number to solve with normal methods. Tell that to


((4!)!!!!!!!!!!)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!/(4!)!!!!!!!!!!.


If you checked that Wikipedia page, by the way, you would see my solution listed as a legitimate method. You're welcome (and I'm sorry.)

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