Response to "Market Scales Puzzle"

When attempting the main question, I tried a whole bunch of other numbers based on some innate interpretation of "understanding" (guess and test), but none of them worked out. It was also both relieving and concerning that the solution is probably unique. On one hand, it's nice that there is probably only one answer, so we can all agree on it -- it's a hint in itself. On the other hand, it feels so restrictive, knowing that only four out of the many integers could possibly work! I also remember getting tangled up thinking about how subtraction is also fair game (placing the weights on the herb side), which made me stumble around. "Would 1 be 1? 3 - 2? 5 - 4?"

In my frustration, I took a detour to the other problem (5 weights, up to 31 grams) since the problem of subtraction wasn't there anymore. I wised up quickly to the fact that 31 is a suspiciously specific number, in addition to being given 5 weights. "Let's give one weight the value of 1 since it's the only way we can get 1. Now we need to get 2. Oh, we need a weight of 2. No other way to do it." I also connected the dots when I saw how many combinations of weights could be possible, so very quickly I realized it was just a binary kind of deal up to 5 digits, so I concluded that the 5 weights should be 1, 2, 4, 8, 16.

I came back to the original question, equipped with the base-wise thinking that helped me solve the problem above... only to fall back to guess and pray. That did not work, so I got desperate and reread the question. As you can see from the picture below, you can see that I used the hints given to guide myself back to base-wise thinking:

I quickly realized the going with base 2 wouldn't be productive, since 4 digits of it would only go up to 15 (1, 2, 4, 8). So I (once again) guessed and tested, this time with base 3. As I went along the target numbers, everything worked out between the base-3 numbers and the constraints of the problem! Thus, my answer to the original problem is 1, 3, 9, 27.

Though I did end up solving the problem with indisputable proof, I will admit that there was a LOT of guessing, testing, and praying. This is something that I hope to address, especially since there is a beauty in terms of bases happening here, along with how they intertwine with the basic addition and subtractions of the problem. As such, my first order of business would be to guide students along the same path that I traversed: undergo some initial immersion in the world of ancient Egyptian mathematics, introduce them to number base systems, and give them the one-pan-scale 5-weights problem first (or even a simpler version where they only need to go up to 15 grams, and only 4 weights are allowed). From this problem, a possible extension after the initial questions could be to generalize. "If we had n weights, what's the highest number of grams we can go while capturing all possible values between 1 and the highest number?" I could ask, with the obligatory request for students to explain, "Why is that?"

Another challenging extension that could be interesting is to lift the restriction that the herbs cannot be used as auxiliary weights. What if you were allowed to use a previously weighed bundle of herbs once? For example, if I had the weights 6g and 4g, I could produce a 2 gram bundle of herbs. From there, I could use the two-pan scale to weigh out an 8 gram bundle (by using the 6g weight and the initial 2g of herbs), and then I cannot repeat this recursive operation again for this instance of weighing. Although it wouldn't be too useful to help students understand (since I'm not even sure if it abides by a pattern of number bases), it could prove to be a fun question to pose to them, to get them thinking mathematically.

To address the last point, this puzzle is definitely driven by my understanding of how base systems work, considering that the solutions to each of the 2 questions were a geometric sequence starting from 1. There's also some interesting patterns that could be gleamed from the solution, if we apply a number theory way of thinking. From 5 to 13, we can notice:

• 9 is in the middle of 5 and 13, and is 4 apart from each.
• The range of 5 and 13 can be created by taking 9, and adding/subtracting some combination of 3 and 1. Possible combinations:
• 0, 0 --> 9
• 0, +1 --> 10
• 0, -1 --> 8
• +3, 0 --> 12
• +3, +1 --> 13
• +3, -1 --> 11
• -3, 0 --> 6
• -3, +1 --> 7
• -3, -1 --> 5
• Observation: there are 9 possible combinations. Each place (+/-/0 3, +/-/0 1) can take a total of 3 states, which matches up with 3^2 (which would be the 2nd place value). The sum of the smaller numbers only goes up to 4, which is also in line.

• Observation: a similar effect can be observed from 27 and the range 14 to 40.
• The largest sum of the smaller numbers is 9 + 3 + 1 = 13, which lines up with the gap between 27 and the range bounds.
• For each of those smaller numbers, there are 3^3 possible combinations of (+/-/0 9, +/-/0 3, +/-/0 1), or 27 possible combinations. The combinations together fill in all numbers 14 to 40.

• Extension idea: If we had 5 weights, what's the highest number we can achieve? Going off the pattern observed above, it would be 3^4 + (3^3 + 3^2 + 3^1 + 3^0), or 81 + 40 = 121.
• If we had n weights, how high can we go? Why is it possible?
• Such extensions aim to build pattern recognition within the context of number theory and base systems.

Of course, there are also questions that can be raised further, like what accommodations we'd need in order to make a base 4 weight system work. Would we need to allow re-weighing as described above? Would we need to allow using each weight number at most twice? These ideas are effectively hypotheses to address the fact that the gaps would be too large (ex. 1 to 4 cannot simply be done with weights 1 and 4; how do we create 2, for instance), and they could serve to guide our extensions as well in order to enrichen student understanding and inquiry into number theory.

September 23, 2024 Exit Slip

"EXERCISE 4.4.2. 

a) Susan's class on mathematics history has students posting on their blogs for various topics and readings, as a way to get her students to engage with the material. One day, Susan's TA Erica wonders, "I wonder how many blog posts will be made throughout the course." If there are 17 students in the course, and Erica estimates that around 20 blog posts are to be written by each student, how many blog posts will be written?

b) Susan can read and evaluate around 5 blog posts per hour while Erica can read and evaluate around 7 blog posts every 2 hours. If the two sit down on the last few days before the term ends to grade every blog post, how long will it take for the two of them to process every blog post?"

This was my attempt at making a word problem by basing the contents and story off a hypothetical sequence of events that my instructor and her TA would go through. Right off the bat, there are some questions to be raised. How would we be sure about our answers if the given information is just in estimates? Why would the professor and her TA wait until the last minute to grade everything, which is not true considering that there will be feedback checkpoints? Of course, the teacher isn't looking for this kind of thought, but instead is simply looking for an answer that is based on the assumption that the estimates are accurate and precise, every premise is valid and sound, and so forth.

Such word problems are within the realm of what I have experienced. Although word problems have mostly phased out by the end of Calculus I in university, every course in mathematics from grade 12 to before had me solving the woes of various constructed entities with problems ranging from the mundane to the utterly bizarre. I am also well aware of the meme that refers to "the guy in the math problems", which is used whenever there is a scenario with outlandish numbers (for example, a picture with a man pushing a shopping cart with 50 boxes of Twinkies at Wal-Mart). While some of these problems are likely applicable to the average student, the fact is that most of them lie in domains foreign to them. In an attempt to make word problems seem "realistic" to a wider audience, they end up feeling contrived and off the mark for almost everyone despite having some plausibility. That being said, I think it's possible to lean into the bizarre nature of some word problems. Why not fully commit to the insanity? Why not make unrealistic premises? I'm also a fan of using names from pop culture and other recognizable sources in word problems, so why not do that too? Although I hope the word problem I presented piqued your interest, I think that such wild word problems could equally pique the interests of students, much like they did for me. A sense of humour is always welcome after all.

Additionally, the concept of a word problem dates back millennia, and in class I've learned that these problems were enjoyed by people all over the world as a form of problem solving, idea sharing, and even riddling one another. Along the Silk Road, cultural melting pots formed at inns where they would share their word problems, sometimes only to flex their mathematical prowess. I think we can gather some inspiration from the enthusiasm of these days and incorporate them into our own classes. Currently, the word problem implementation in textbooks and classrooms is vaccine-like. We set up sterile targets for students to shoot down, and they are, again, contrived. There is a missing element of creativity that we can introduce. Why not encourage students to make their own word problems out of the stories and struggles in their lives? I could see myself setting aside time for students to form and exchange their word problems in class, which could be beneficial in terms of promoting creativity, collaboration, and most importantly: application to real life that the students would care about. I do want to note that it's okay if the questions they come up with seem contrived; even then, there's a fun storytelling element that allows the mathematics to embrace a written art form, whether fictional or non-fictional. Thus, I also believe that by allowing students to write their own mathematical stories, we can introduce an artistic flair that would pique many interests. In short, I would like to use word problems in my own teaching to promote collaboration, artistic and historical thinking, and the standard intents of practicality and realism.

To address the second question on the prompt for this blog post, the first thing that comes to my mind is simply bringing awareness to the history of word problems! Many students likely haven't even considered that there's a rich history to word problems, instead believing that they are a psyop intended to confuse and torture them under the guise of "real world connection". I'm assuming that mathematicians, math teachers and historians have compiled a bank of word problems throughout the ages -- I understand that they are already being used in our textbooks, and we can continue using them, except this time we'd acknowledge where, who, and when they came from, or what the problem was used for. My belief is that kids generally love harmless fun facts, and with enough repeated sightings of things like names, eras, civilizations, etc., they will start to recognize the rich history behind the problems they are solving. 

Another idea is to incorporate assignments where students simply search for historical word problems and inquire about them on their own, with a list of resources and starting points provided, of course. In these assignments, they could happen once a sufficient number of concepts have been taught such that students have a greater scope to find a relevant question from. Although I haven't gotten too deep into working on Assignment 1 for this course yet, I suspect that such an assignment (give starting points, ask students to solve their question in modern and ancient ways to build appreciation for both, and extend) could be replicated in a high school classroom. I look forward to hearing about more ideas on how to connect with the history of these word problems to our students, and hope that I was able to offer some fun ideas of my own! Even though I struggled through them in high school, I recognize their importance and do not wish for them to be thrown away, ending a tradition of millennia.

Response to "Mathematics in ancient Egypt"

From the article, it confirmed the in-class discussions we had on the mural, which I appreciate. The mural depicts ancient Egyptian surveyors measuring agricultural land, most likely for the purposes of fair distribution after the annual floods. The article also discusses the many units of length, and how angles were measured.

My questions correspond to these units of length, and the methods of constructing angles. The first oddity that I thought about was related to the differences between the "royal cubit" and the "short cubit". It made sense to me that the ancient Egyptians couldn't settle on a formal definition considering that it took forever for places like Europe as well, but the article seems to default to the royal definition of 7 palms and 28 fingers, never mentioning short cubits again (perhaps for the purpose of simplifying cubits to the reader). Another strange thing about the topic of cubits is that it was considered "the length from the elbow to the tip of the middle finger". I asked myself, "Who was the standard reference for the exact length of a cubit?" This is further mystified by the article's wording; it simply says "most often about 524 mm" instead of something more concrete. Although not the same, it feels like it has tie-ins with the imperial measurement system where units are named after tangible objects such as yards and feet. ("What do you mean my foot isn't 12 inches long? Is anyone's foot 12 inches long?" I would ask as a child.) 

The most surprising part is that despite all these inconsistencies and seemingly arbitrary details, there are genuinely beautiful results that come from the cubit, such as the remen and its properties. The remen was a unit of length that was half the diagonal of a square royal cubit. Having a square of length 1 cubit, it has 2 times the area of a square of length 1 remen and 1/2 the area of a square of length 2 remens, and all of this seems too good to be true, especially since the cubit's length is stemmed from a seemingly random ancient Egyptian's bodily parts and no one could really settle on its actual length.

The other source of confusion I initially had was with the section on constructing right angles. Right angles seemed to me like they should be easy to construct, but this is also coming from someone who has never needed to measure or survey large areas and lengths. That being said, my question was, "Is it that hard to construct right angles, even if the distances are that large?" I was reading the article and they gave explanations involving all this equipment and equilateral triangle and rhomboid strategies, and thought, "If I just showed them what a right triangle is, why can't they just mark a perpendicular line off whatever base line they are going off of?" I could see different angles needing different techniques, but a right angle doesn't seem like the hardest thing to create accurately as long as we keep our ropes straight and angle maintained at the intersection point of the two perpendicular lines. I can see the plumb bobs being useful if we had to construct a right angle vertically, but with a flat plane it just seems excessively elaborate. One reason for all these elaborate methods that I could think of is this: they may also need to verify if an angle is right or not. Maybe the ancient Egyptians got interrupted during angle construction and they had to revisit their measurements, in which case these elaborate methods would be worth it.

Ancient Egyptian multiplication and division

Here I attempt to perform 3 divisions using Ancient Egyptian techniques.

The first one is taken directly from the original blog post as an example to learn from. The algorithm goes:

1. Set 1 in column one and the denominator in column two.
2. For subsequent rows, double the column values from above.
3. Stop doubling if the next column two value will be greater than the numerator.
4. For the next row, halve the column values from the first row.
5. Continue this halving until an odd number is reached.
6. Find the numbers in column two that add to the numerator. Take the corresponding column one numbers and add them to get the answer. 

This is done in the bottom-left example that I tried out myself.

On the top-right, I decided to explore the case of an odd whole number divisor. Where I asked "is this legal?", I was unsure if it was allowed within the bounds of the Ancient Egyptian techniques. These uncertain steps were: not halving, but dividing by the denominator, and doubling the 1/3 row's values to get (2/3, 2).

Within the limits of my algorithm, however, those moves should not be legal. The 1/3 step may have been safe since it's in the 1/x form, but the 2/3 step may not be allowed (despite it being 2/3, the way I got 2/3 would not be transferrable if the denominator were 5).

As such, I will do 14/5 using Ancient Egyptian techniques, with some potential overstepping of the algorithm:

So it turns out this gets the right answer (making a 1/x row, then iteratively doubling it). If we were doing 12/5, then we wouldn't make the 4/5 row. I also want to note that instead of going from 1/5 to 1/25, I doubled 1/5. (If I did 1/25, the corresponding row would be 1/2, which wouldn't be particularly productive here.) This should work because if we start with a 1 on column two and iteratively double it, we can cover all numbers just like with repeated doublings starting from 1 in the first column.

If describing the "technique" as purely 1/x fractions and doubling/halving, I don't believe that the technique works with all whole number divisors, due to some cases of odd divisors being indivisible by 2. If we incorporated other unit fractions, it doesn't appear to be possible based on my definition of technique unless we break the doubling/halving rules like I did with the 14/5 example.

Regardless, I do want to know how the Ancient Egyptians handled odd number divisor cases. Until then, I will not be spoiling myself.

Response to "Babylonian Algebra"

When I went through this reading, I felt like there was a relatability here even across the millennia. While I was struggling in my university math classes and using roundabout, borderline brute force yet creative strategies, the Babylonians were working through their word problems with a similar level of grit. As such, I continue to have respect for these ancient mathematicians, going into hard mathematical battles with nothing but a long spear and broad rocks (see what I did there?)

Thinking about it, there were still mathematical principles at play in these pre-algebraic times. Linear, quadratic, and higher order equations were still there to be solved, after all. The Babylonian mathematicians in this reading gave some examples of how one would do this, which I could sum up as plenty of formula and table of values usage. For example, the solution to (what is basically) ax = c -- they had a formula ready to use, and a table of reciprocals. I imagine that the wording would be arcane for these things (I had to reread the word problem descriptions multiple times to even understand), but with the use of geometric terms like "length", "width", "volume", "area", and "volume", even though it wasn't declaratively algebra, they had something to work with along the lines of an algebra.

Which gets to the next point: I believe that at the core, mathematics is mostly about generalization and abstraction. Throughout history, we have tried to simplify mathematical processes and solutions by proving and deriving things like rules and defining terms to describe mathematical entities. In the reading, we see many instances of this with formulas, such as the solution to ax = c or the quadratic solutions, or unknowns being defined with the list I quote-marked above. Even these stem from lower level abstractions; multiplication, for example, can be seen as repeated addition. Exponentiation comes with many laws to address things like integers and fractional powers and bases. With number theory, I remember learning about the Chinese Remainder Theorem. We had formulas to solve systems of congruences for many different cases, which stemmed from relational experimentation in order to give us abstracted instruments. All of this talk reminds me of the computer science concept of abstraction, where complex details are "abstracted away" in order to give users and programmers easy-to-use instruments.

Of course, there are some parts of mathematics that cannot simply be generalized or abstracted. Even with the Chinese Remainder Theorem, it had to be split up into various cases which reduced its generalization. There are also atomic principles such as 1 + 1 = 2, which doesn't lend itself much to abstraction. Regardless, it cannot be overstated how important generalization and abstraction are to the field, as they are at the core of foundation building that allows mathematics to grow so much while abiding by a universal sense of logic.

As such, I find it a bit harrowing to state general/abstract relationships without algebra. So many of our definitions and formulas rely on the symbols of algebra, so to write down something like the quadratic equation's solutions in the following manner would be cumbersome and excessively verbose: "First, you get the area of a square with the breadth given. Take away four of the area formed by the given length and height. Then, get the square root (and so on. This was certainly difficult, as I caught myself saying "let's call this square root n"!)". If I think about it, it would still be possible to describe these relationships with language, but it would just be unnecessarily verbose and hard to decipher. We would still be able to use visuals, which I surmise the Babylonians used considering their affinity for geometrical terms. Visuals are already a key part of understanding things like geometry (showing shapes and their dimensions), calculus (graphs of functions) and especially graph theory (showing vertices and edges, and reasoning with them visually), though I suspect that with visuals we'll still say "this" and "that" a lot when describing the relationships. Regardless, I can definitely say after struggling through the article and writing this that algebra is a powerful tool that I cannot be more thankful for. Long live algebraic notation, and may it continue to transcend cultural and language barriers as a universal script.

Response to two resources on Babylonian-based history of time calculations, base 60 and base 12

In the two articles, they discuss the possible origins of base 60 in ancient times. The first article, however, spent far more time not only discussing, but also giving validity to various theories of the origins of base 60. This was one big point of inconsistency between the two articles, as the second article was more dismissive of the elaborate, "scholarly" theories of base 60's origins. The reasoning for this was pretty funny because it essentially asked, "Did the ancient Sumerians hold a committee to decide a base? No!" While funny, it did raise a fair point as the author of that article decided to take the naturalist route, coming up with everyday finger-counting theories that sounded quite convincing, or a theory involving the marriage of a base 5 and base 12 society.

Regarding the ways of measuring time throughout the ages, the first article shone a ray of light on the Egyptians, who in turn made sundials to interpret time using said light. Just kidding. However, their sundial and astronomy advancements were nothing to scoff at, as it resulted in our 24-hour, two parts 12-hour day. They struggled with the boundaries of these realms due to their dependence on sunrise/sunset, which resulted in complications in defining hour lengths. Personally, I can relate. During the summer when I was spending hours on a game, but I had a very consistent daily routine. The day was basically defined by four periods of play (well, free time in general), and so I split the day according to the following structure (red = game time - I will admit, the summer was not particularly productive):

Now that I'm back in school, this entire view of the day has been completely thrown out as I readapt to a more weekly-consistent (versus daily-consistent) life. I'm still in the process of figuring out my image of a week, but the reset point is looking like Wednesday, which is my longest day and also the day that weekly tasks reset in my game. Friday on the other hand is the slightly more relaxing part of the week, at least right now.

On the scope of the year, there were some theories in the second article where the year had 360 days, which was a multiple of 60. Of course, this was also dismissed due to the author being "certain" that the Sumerians knew the year was longer than 360 days. For me though, while I still abide by the 365 day Gregorian calendar, most of how I geometrize the year is from my seasonal anime watching hobby. Instead of the seasonal splits halfway through four of the months, I could tell when a season was ending or starting. My "currently watching" list updates reflected this - adding shows meant a season was starting, and vice versa. However, there are very few commonalities except for within the specific season and year, so my structure ends up being an endless arrow of time, divided into 3-month segments titled by their season and year and defined by the shows that I watched:

To address the surprise part: I was mainly surprised at the ingenuity of the ancient peoples, especially with the ancient Egyptians. I have to credit them for giving me the 12/24-hour splitting of the day that allowed my own geometry to flourish, and for their creativity in using the stars to get to that point. However, I can safely say that the point of no return for me was the 14th century when all the 12/24 and 60 minute/second splits were well-established, and the clock as we know it was born. I do not wish to use the stars or a sundial since clocks are far easier, though if I ever find myself stuck in the wilderness, I'll know where to turn to: a long stick and the sun, or the cosmos above without the light pollution of the cities.

----------

The anime in the last picture were chosen based on their relative popularity and/or my own preferences. However, I have watched all of them when they were airing. For the curious, their names are:

Fall 2022: Bocchi the Rock! / Blue Lock

Winter 2023: Bleach: Thousand Year Blood War / Vinland Saga Season 2

Spring 2023: Hell's Paradise / Mashle

Summer 2023: Jujutsu Kaisen Season 2 / Zom 100

Fall 2023: Frieren: Beyond Journey's End / Spy X Family Season 2

Winter 2024: A Sign of Affection / Solo Levelling

Spring 2024: Kaiju No. 8 / Wind Breaker

Summer 2024: Oshi No Ko Season 2 / Alya Sometimes Hides Her Feelings in Russian

Be mindful that some are sequel seasons, so you may have to watch whatever came before first.

Why Base 60?

A little exploration of Babylonian numbers

Presently, people are most familiar with using a completely base ten system (not saying "10" here, because that could mean anything depending on the base you choose... just kidding). It's great - we have 10 fingers, 10 toes, and are all 10 out of 10 individuals, I think.

However, the Babylonians had a different idea - a base 60. The places went something along the lines of (right to left, starting from the decimal comma):

• 1, 0 to 59 (the unit place; note, within this place there are two places for two different types of wedges)
• 60, 0 to 59 (multiply by 60)
• 3600, 0 to 59 (multiply by 60)
• ...and so on (multiply by 60 for the next place)

If we used our Arabic numerals, then something like 337 using this base system would be:

(3 x 3600) + (3 x 60) + (7 x 1) = 10987 in our regular base 10. Interestingly, the Babylonian system would use far fewer digits than ours.

So let's ask: why would they use 60 for their places? When do we use 60?

First, we will speculate. There were two trains of thought that I had: the first being its divisibility, and the second being its relation to the human form. Regarding the former, 60 is a highly composite number that can be divided by 2, 3, 4, 5, 6, 10, 12, 15, and 30. Fantastic! There was a point brought up in class as well regarding clocks which use 60 for minutes and seconds, and I find it particularly convenient due to its divisible nature. Mathematically, this would make sense. From the factors, we can also make connections with the number 12, which is the number of months or Chinese zodiac animals, or the number of astrology signs. Meanwhile, the number 10 is strictly divisible by only 2 and 5. Not as flexible!

On the other hand, we have 5 fingers. Well, the first hand probably also had 5 fingers, but let's not get into those details. In any case, these fingers can be convenient for counting. Children learn to count to ten on their hands as they have 10 fingers, whereas computer scientists get clever and count to 1023 using a binary system (fingers can be one of two states, after all). Maybe they'd get extra creative and use their tongue to represent the 1024s place, and count up to 2047. But I digress. In the case of the Babylonians, I suspect that they used their digits on one hand to tally up the 10s in the sub-10s place. Both hands would be used if they wanted to work on the units sub-place.

Other speculations could be that 60 represented something special for them. The people of that time and place were really into the moon, and a moon cycle is about 30 days which is half of 60. Maybe a Babylonian ruler really liked the number 60, but at this point the theories are becoming outlandish.

After some quick research, what have I found out about the significance of 60?

Upon doing a quick search on the significance of 60, I found that I was on to a lot of things (such as the divisibility and time aspects) but also omitted various others! The key takeaway for me is that this seemingly arbitrary number actually has a solid grounding in our civilization, from ancient to contemporary times.

Before the sexagesimal system was conceived, the prevailing theory started with two peoples with different base systems eventually merging to become one group. The original base systems were 5 and 12, so the marriage of the peoples led to the marriage of their base systems of 60. From there, the ancient peoples made advancements in measuring time using the number 60, which had ramifications on today's time-telling. Do you see those modern SI units where every unit is some power of 10 up or down from another? No one is casually calling a thousand seconds a kilosecond unless they are in a specific discipline that requires it. On the topic of keeping track of time, there are also ties to Indian and Chinese calendars, with a 60-day unit of time being particularly popular. Amongst the Chinese, their beliefs in the five elements and twelve zodiacs have parallels with the marriage depicted above, as the Chinese also believe in a full 60-year cycle that features all possible pairings of elements and zodiacs.

Other significances of 60 that can be found in modern times include: degrees as a unit of angle (360 degrees makes a full round, and 360 is divisible by 60, which is further divisible by all the factors above - very helpful especially when dealing with trigonometric functions and the unit circle), electrical applications (for example, monitors and programs running at "60 frames per second or hertz" - as someone who plays fast-paced video games, the subdivision of 60 frames in a minute is of personal significance when comparing action speeds and input delay issues), and even the latitudes and longitudes that act as navigational grids (180 of each, divisible by 60; smaller units also abide by a sexagesimal system). The former two items are particularly applicable to me, given my time spent working with math and gaming, so I have to give a shoutout to the Babylonians for having the right idea. However, I imagine the people like the navigators of the world are also thankful for the divisibility of 60 as they interpret their coordinates.

Sources of information and inspiration:

• https://thoughtco.com/why-we-still-use-babylonian-mathematics-116679
• https://ridingthebeast.com/numbers/nu60.php
• https://en.wikipedia.org/wiki/60_(number); particularly the ideas under "In other fields"