# Dr. Stone - Arithmetic Battle 2020 October 26

So I’m watching Dr. Stone, and in the science demonstrating contest between Senkū and Chrome they have a final battle over who knows more math. This is a list of some of the math I think Senkū would have used to crush Chrome or at least some math tricks I think are really cool. I know that it doesn’t go down like this, but I like to think it could have possibly went down a couple different ways and might have hit on these. It’s obvious in the manga that Chrome only knows Arithmetic, but these are some cool tricks that could’ve been applied.

# Gauss Sums

There’s this story about how Karl Gauss (a famous mathematician) out smarted his teacher and found a simple way to sum up the numbers from 1 to 100. Read the full story for a better telling than I’m capable of, but it would have been cool for Senkū to have brought up if Chrome had asked about any kind of summation of series, however since he barely knows addition this probably would not have come up. Here’s the formula if you want to investigate it further and the Wikipedia page.

$1 + 2 + 3 + 4 + \dots + N = \sum^{N}_{k=1}\frac{n(n-1)}{2}$

# Polynomials & Calculating Airtime.

One of the coolest things in introductory physics is calculating airtime and a projectile’s y displacement over time. It’s super handy and would be useful for any aspiring scientist to know. It allows us to know important things like where a projectile will hit or how deep a hole is. All useful for anybody doing any kind of mapping or engineering. I’m not going to provide an example here, because it’s out of scope, but go look at any basic high school physics class for example. It’s based on a polynomial, $\delta{x} = \frac{1}{2}at^2 + vt$. 2nd-order polynomials have a super easy solution that most people can use called the quadratic formula.

## How it might have gone down.

Senkū challenges Chrome to determine the airtime of a ball thrown up into the air, and Chrome is speechless without an answer. Senkū then goes on to explain how to determine the fundamental acceleration due to gravity, and explains what airtime is. This of course assumes that Chrome even understands what a second is and units to measure distance.

How would a stone age civilization even have those units? How would Senkū even connect our modern units to those in this freaky stone age future? The Egyptians had units defined by their king’s body parts. In medieval times it was defined by the monarch’s body parts. It was only relatively recently that units of length were standardized, and time wasn’t really accurately measured until the advent of the mechanical clock or at least on the level of a second until about the 1400s with pendulum clocks. So maybe Senkū would connect the consistent period of a pendulum to time. A whole bunch of questions that were not answered by the anime.

# Different Base Arithmetic

What if Chrome had been doing his Arithmetic in a different base, say something like base 8 or base 20. Many civilizations have used different base number systems in the past. Babylon, one of the first civilizations, actually used a base-60 counting system, as an aside at one point this system was supported in YAML.

My entire point behind is this is what if something similar to this comic happened with Senkū and Chrome. Chrome states that something weird like “7 + 7 = 16,”, this is not true in base-10, but it is true in base-8 or octal.

# Magic 9

Magic 9 is a weird rule that I learned in 2nd grade about subtracting 9 from numbers less than 20, but greater than 9. It’s mildly interesting. You interpret the digits of the larger number as individual single numbers and add them together. The resulting sum is the answer of the original subtraction. Proof below.

19 - 9 = 10 = 1 + 9
18 - 9 =  9 = 1 + 8
17 - 9 =  8 = 1 + 7
16 - 9 =  7 = 1 + 6
15 - 9 =  6 = 1 + 5
14 - 9 =  5 = 1 + 4
13 - 9 =  4 = 1 + 3
12 - 9 =  3 = 1 + 2
11 - 9 =  2 = 1 + 1
10 - 9 =  1 = 1 + 0


Overall not that useful in the real world but interesting. It might only be applicable to base-10, but there’s room to explore with it.

## How it might have went down

Honestly, it probably would not have come up, but I wanted to write about this on my website, and now I have.

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Written by Henry J Schmale on 2020 October 26