Magic Square

Initially, I subtracted 9 from 15 to see the lowest number two others had to add to. Thus, I was able to see that 1,2; 1,2; 1,4; and 2,3 were not able to be in the same row, column, or diagonal. After this, I found that the only options for 1 are 1 + 5 + 9 and 1 + 6 + 8, meaning 1 couldn't be along the diagonal (which would require for at least 3 options). Initially, I put 1 in the second column of the first row:

Then, I tried placing 5, 9, 6, 8, in in the same row and column as 1 without much reasoning. After that, I filled in the rest of the box so that rows and columns added 15. However, I forgot to account for diagonals and ended up with 9 and 6 in the same column, which meant the sum was over 15. Then, I realized since 9,6; 9, 7 or 9, 8 couldn't be in the same column, row, or diagonal, when I arranged 5, 9, 6, 8 around 1, the only possible number that could fit in the middle was 5 (since 6 and 8 would end up in the same diagonal or column as 9 otherwise). That meant 9 had to go in the same column as 1 and 5, and I placed 6 and 8 in the same row as 1:

From here, I filled in the remaining diagonals, then the remaining two spaces to get my final answer:

I realized that this solution wasn't the only organization of the square because I made some arbitrary choices (i.e. putting 1 in the first row, second column instead of the 2nd row's first or third columns, or the last row's second column, and when placing 6 or 8 around 1).

Eye of Horus and Special Numbers

Image source: http://blog.candere.com/wp-content/uploads/2017/09/Navratri-in-Gujarat.jpg

One of the interesting things I found about the Eye of Horus was the adding the 6 parts together was 1/64 away from being one. In one source I found, they stated that the leftover said to be attributed to magic or that nothing is perfect. Considering the Eye of Horus's mythological association, I think it's more likely that the magical explanation was used, but the notion that nothing is perfect is quite wise and a poetic thought as well. Another source talked about how it is the first known example of a geometric series which was pretty interesting. It would make a good introduction to the concept if taught in a classroom.

The only two numbers with a story I could think of off the top of my head were 0 and Pi but in my search, I also found numbers like 666, which is known for its association with the Devil in Biblical texts. However, I was not aware that an older text was found stating the number associated with the beast was actually 616 (see https://en.wikipedia.org/wiki/Number_of_the_beast). The translation hasn't been completely agreed upon (https://en.wikipedia.org/wiki/Papyrus_115) so it isn't a fully accepted association at the moment. I wonder if the switch over will occur in the case that older texts are found and confirm this. It is such a culturally ingrained association that it would likely be difficult to do so.

Coincidentally, a festival called Navratri (which translates to nine nights) started today. The Hindu religious story associated with the festival is that it is a celebration of the goddess Durga defeating the demon Mahishasura. The nine days represent nine forms or avatars of the goddess and people often do some form of fasting during this time. I learned about a new concept this year that some regions set up dolls on multi-tiered pedestals but the thing that struck me about this was that there could only be an odd number of steps in the display. I wasn't able to find a reasoning for this but some sources said the acceptable number of steps are 1, 3, 5, 7, or 9, while others said 3, 5, 7, 9, and 11. I called my grandma to see if she had any insight but she said she didn't know. However, she said that 9 is an auspicious number in other Hindu religious stories as well so the choices are probably a reflection of which are closest in property to 9. For example, 9 is odd so 1, 3, 5, 7, and 11 are closer in property. Additionally, she pointed out that 9 multiplied by 1, 3, 5,  or 7 results in digits that add directly to 9 but 11 doesn't fit that pattern so that might be why some people exclude it. Others might include it because 99 is actually two nines so possibly double the good luck (and 999 (111 steps) would be excessive... ). However, these are just our guesses.

Was Pythagoras Chinese?

Image source: https://upload.wikimedia.org/wikipedia/commons/c/c3/Chinese_pythagoras.jpg

I think it is important to acknowledge non-European mathematicians not only so that students see representations of different cultures, but also to give them a clearer sense of how math developed. It's important to show them that there wasn't just one group of people building on work that was solely from their culture, which is a misconception that can easily come about when we only focus on prominent European mathematicians. In addition to this, learning about how different cultures had different prominent ancient texts (e.g. Euclid in Europe, the Jiu Zhang in China etc.) shows how advance math was being worked on for years. Seeing the different cultural contexts attached to the texts makes learning about them feel like an important look at human history and can make students feel more connected to what they're learning about. There is a lot of depth we miss out on by simply telling our students a concept is named after a mathematician or group of people but not talking about those people.

While I believe names can hold significance, I think it's trickier when it comes to older theorems and concepts where it is difficult to pinpoint who discovered it "first." That being said, I think it is better to name concepts and theorems after people to retain the connection to humanity and culture while thinking about math. If the Pythagorean theorem was called the Right Triangle Theorem, it might be easier to remember what it was for within the mathematical realm, but it makes the theorem faceless. Also, as new theorems arose, naming would become considerably harder to the point where theorems and concepts might just become "Right Triangle Theorem 1" and "Right Triangle Theorem 2" etc. In terms of accurately giving recognition to the proper mathematician, I think it has become easier to accurately identify who was the first to prove newer developments. Even if mistakes have been made, there is more evidence and source material available to rectify those mistakes. However, even if there is speculation about Pythagoras being Chinese or other cultures accurately proving the theorem before him, I'm not sure if changing the name would be useful. Instead, I think it would be better to focus on talking about those other cultures and texts, and recognizing that regardless of who ultimately proved a theorem or concept, it is the work of humans and an important part of our development as a species - not just something that belongs to one culture.

The Method of False Position

Image source: hebbarskitchen.com

Question:

My grandma uses a household bowl to measure ingredients for most of her dishes. While making khichdi, she usually puts one full dish and its eighth of dal into an empty pot, which brings it to the 500 ml mark. I need to find the measurement of the bowl so that I can figure out her reference point for future recipes. What is the bowl's measurement in millilitres? 

Solution:

x + ^x⁄₈ = 500 

Try x = 80: 

80 + ⁸⁰⁄₈ = 90 

90 multiplied by what gives 500? 

90        1 

180      2

360      4 

720      8 

 45       ¹⁄₂

 15        ¹⁄₆

  5         1⁄₁₈

360 + 90 + 45 + 5 = 500 so we have 4 + 1 + 1⁄₂ + 1⁄₁₈ = 50⁄₉ multiplied by 90 gives 500. 

Thus x = 80 x 5 5⁄₉

80         1

160       2   

320       4

40         1⁄₂

20         1⁄₄

10         1⁄₈

 x = 80 + 320 + (80 + 320)/9

x = 400 + 400/9

9       1 

18     2

36     4 

72     8 

144    16

288   32

576    64

x = 400 + 32 + 8 + 4 +  4/9 = 444 4⁄₉  ml

History of Babylonian Word Problems

Image source: https://www.open.edu/openlearn/sites/www.open.edu.openlearn/files/imported/14056/cuneiform_catland.jpg

It's interesting to see that Babylonians had both math application questions and pure math ideas presented in application question form. While the applied math questions were for practical purposes, the "pure" math versions were just because they could. I think that ties into how humans have an urge to test boundaries. I was in an E.A. in a Family Studies course once and the teacher explained how the problem with constantly keeping children (or anyone, really) busy is that they don't have an opportunity to be bored. Boredom, she argued, is essential because it facilitates creativity. I'm sure after drilling an abundance of similar practical math exercises, someone eventually became bored enough to wonder how they could push their boundaries and find a new way to engage with math. 

As to practicality and abstraction with relation to contemporary algebra, I partially agree with Jacob Klein's notion that only modern algebraic notation qualifies as "purity of form"(p. 8). I think the clearest way to make generalizations in abstract math is through modern algebraic notation but I also recognize that it's the convention that I'm most familiar with and that there may be other representations out there that I am not aware of. In the case of the Babylonians, I can see how their pure mathematics wasn't deemed to have a pure form because the proposed problems were based off of real world applications, but I also believe that it doesn't make that math any less valuable. In my opinion, purity of form is a technicality and the substance of a pure math problem is more useful. I believe venturing into the realm of pure math as the Babylonians did was important progress in the field of math. It set the stage to continue pushing the boundaries of the use of math and created the foundation for what came afterwards.

Assignment 1 Solution, Extension, and Reflection

My group's slides can be found by following this link to our slides: https://docs.google.com/presentation/d/1v7p19i8AY7yPETjULVteIXuK2wauLVCQ3dvWLmrbIYs/edit?usp=sharing

Edit: Reflection post-presentation

In creating this project, we really had to put ourselves in the place of Babylonians years ago, which wasn't exactly easy. I came across examples of their architecture that contained archways, which put me on a track to hypothesize that sagitta and chord measurements were needed to complete the arc calculation. However, I wasn't able to find concrete evidence to support this, and actually had to research modern calculations of archways before I felt more confident about my guess. It would be interesting to see if there were other applications of the sagitta and chord in Babylonian society. I believe someone guessed during our presentation that they might have been used in wheel calculations. Although I wasn't able to find any evidence to claim this through my limited Google searches, it would be interesting to learn more about how Babylonians came up with the design of the wheel and if they did make use of these measurements. On a personal level, I really enjoyed researching Babylonian culture and architecture. The art on the Ishtar Gate was beautiful and it is amazing to know that these impressive structures were created so long ago.