Market Scales Puzzle

Let's say we use 1, 2, 3, 4g of weights to measure

Left: Herb 1 Right: 1

Left: Herb 2 Right: 2

Left: Herb 3 Right: 3 or 1 + 2

Left: Herb 4 Right: 4 or 1 + 3

Left: Herb 5 Right: 2 + 3 or 1 + 4

Left: Herb 6 Right: 4 + 2

Left: Herb 7 Right: 4 + 3

Left: Herb 8 Right: 1 + 3 + 4

Left: Herb 9 Right: 2 + 3 + 4

Left: Herb 10 Right: 1 + 2 + 3 + 4

It seems we have used up all the weights and we're stuck. Let's try a different approach.




Let's say we use 1, 3, 6, 10g for weights

Left: Herb 1 Right: 1

Left: Herb 2 + 1g Right: 3

Left: Herb 3 Right: 3

Left: Herb 4 Right: 3 + 1

Left: Herb 5 + 1 Right: 6

Left: Herb 6 Right: 6

Left: Herb 7 + 3g Right: 10

Left: Herb 8 + 1g Right: 6 + 3

Left: Herb 9 + 1g Right: 10

...

Left: Herb 20 Right: 10 + 6 + 3 + 1

Left: Herb 21 Right: X

We have used up all 4 numbers for weights, even if we add all four numbers together, the heaviest weight we will get is 20g, so we're stuck again.


Let's do the other way around this time, starting with the total weight that gives us 40. Since we need 4 numbers that would give us the combination of 40g for the heaviest weight, we can safely eliminate the weight of 35 to 40g.

Let's start with weights of 1, 2, 3, 34g.

Left: 34 + 1 + 2 + 3 Right: 40

Left: 34 + 2 + 3 Right: 39

Left: 34 + 1 + 2 Right: 37

Left: 34 + 2 Right: 36

Left: 34 + 1 Right: 35

Left: 34 Right: 34

Left: 33 + 1 Right: 34

Left: 32 + 2 Right: 34

...

Left: 28 + 3 + 2 + 1 Right: 34

Left: X Right: 34

We can quickly see that we ran out of options for giving us weights lower than 28g.

What about 33g? The combination we can get from 40g is 1, 2, 4, 33g. However, we ran out of options when we get to weights lower than 26.

Left: 26 + 1 + 2 + 4 Right: 33

Left: X Right: 33


I find that when we take away the heaviest weight out of the four, the herb plus the remaining three smaller weights added together must be larger or equal to the biggest weight. So we must figure out a way to form the combination of the remaining smaller weights plus the herb added together to be equal to or larger than then heaviest weight, or the remaining 3 smaller weights can form different combinations to find the weight of the remaining herbs.

Let's take a leap and jump straight to a lower number, say 28. Let's use the combination of 1, 2, 9, 28.

However, we're stuck when we get to herb equal to 15 grams because:

Left: Herb 15 + Any weight combination Right: X

We also know that 9 + 2 + 1 = 12, so we can see that we can have herbs up to 12 grams.

The other combinations including 28grams couldn't work either.

But it seems we are getting close!


Let's try a lower number 27. Say we use the combination of 27 with 3 numbers that gives us 40 is 1, 3 and 9.


Left: Herb 1 Right: 1

Left: Herb 2 + 1g Right: 3

Left: Herb 3 Right: 3



Left: Herb 4 Right: 3 + 1

Left: Herb 5 + 1 + 3 Right: 9

Left: Herb 6 + 3 Right: 9

Left: Herb 7 + 3 Right: 9 + 1

...

Left: Herb 13 Right: 9 + 3 + 1

Left: Herb 14 + 9 + 3 + 1 Right: 27

Left: Herb 15 + 9 + 3 Right: 27

Left: Herb 16 + 9 + 3 Right: 27 + 1

...

Left: Herb 39 Right: 3 + 9 + 27

Left: Herb 40 Right: 1 + 3 + 9 + 27




So it seems like we have our answer in 1g, 3g, 9g and 27g.

This confirms my theory, that in order for the scale to work without gaps, the sum of the 3 smaller weights with the herb has to be equal to or larger than the heaviest weight. The smaller weights can also form different combinations with the herbs to find their respective weights. Essentially, the smaller weights have to fill the gaps between lighter herbs and the heaviest weight.


There were other combinations such as 1, 2, 10, 27, but the numbers don't add up to the weight of some herbs, or we encounter a dead end when calculating.




What if it was a one-pan scale?

Let's say this time, we only have a one-pan scale. We need to find 5 total weights that give us 31 grams together.

Let's say we use 1, 2, 3, 4, 21. We can quickly see that this doesn't work because we can't make a combination between 11 and 20.

Since we know that we 1 + 2 = 3, we can safely disregard 3 as one of our numbers and use the next number 4 as one of the 5 weights. Let's try 1, 2, 4, 8, 16. The reason we do this is because we can make every number weights from 1 to 7 through numbers 1, 2 and 4, therefore we can jump straight to the next number after 7, which is 8 in this case.


Herb 1 = 1

Herb 2 = 2

Herb 3 = 1 + 2

Herb 4 = 4

Herb 5 = 1 + 4

Herb 6 = 2 + 4

Herb 7 = 2 + 4 + 1

Herb 8 = 8

Herb 9 = 1 + 8

Herb 10 = 2 + 8

...

Herb 15 = 1 + 2 + 4 + 8

Herb 16 = 16

...

Herb 31 = 1 + 2 + 4 + 8 + 16


We know this works because we can use number 1, 2, 4 and 8 to add up to any numbers up between 1 to 15, which also means as long as the numbers we're trying to make is 15 or less it will work, which is why 1, 2, 4, 8 and 16 works, because numbers between 16 and 31 is 15 or less.

One interesting fact I noticed is that 1, 2, 4, 8, 16 are all results from powers of 2:

2^0 = 1

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

Similarly, for the two pan scale questions, numbers 1, 3, 9, 27 are also results from powers of 3:

3^0 = 1

3^1 = 3

3^2 = 9

3^3 = 27




By working with this puzzle, students will develop logical thinking by using a set of given numbers, starting with 1, and finding the combinations that would eventually lead us to the correct weight. Students can be asked "What's the next number that fits the combination after 1?" This encourages them to dive deep and explore the different combinations they can create. The puzzle also allows students to understand how pan balance scale works, while they develop their thinking and problem solving skills.




This puzzle connects in a way where I'm already familiar with both how the two-pan scales and the one-pan scales work. We have to try different combinations of weight in order to find the correct total that involves working with logical thinking. On the other hand, the patterns that I discovered from the puzzle are also related to exponents. This also connects to the idea of different number bases where numbers are represented with powers of 2 and 3, which is quite fascinating!

More on Babylonian word problems

While reading the article, I came to the realization that math word problems have a very long history. I was amazed to learn that some of the earliest word problems existed during the early Babylonian period. At the same time, the article raised concerns about the practical purpose of the word problems, which made me wonder what their original purpose was and whether the word problems we see today developed from the Babylonians.

One thing that really stood out to me was Robson's discussion in the article about how these problems taught practical skills. On the other had, many of the skills taught through these problems are no longer relatable to real life. This made me wonder how impactful these skills were historically and whether they can still be applied to modern society. When we think about word problems today, we often picture lengthy questions that we complete for an assignment and then forget about them after class. We rarely stop to consider how the math concepts within the problems could be applicable to our daily lives. During early periods, the Babylonians were able to use math problems to deal with agricultural and commercial administration. If we can find ways to connect the word problems to real life, perhaps they could become more useful to students.

After reading the article, one question I have is: How do we determine whether the word problems we use are impactful? As educators, we are always encouraging students to succeed, and if we are going to implement these problems in classrooms, I believe that prioritizing problems that connect to real life situations could make learning more meaningful.

Did Mesopotamian scribes have algebra?

Upon reading this article regarding Babylonian algebra, I find it very intriguing how at a time before algebra was invented, the Babylonians somehow found a way to develop sophisticated numerical methods for solving various forms of equations. People could express math relationships through repeated trial and error of equations and imaginative use of various tables at the time. To add on, by using a sexagesimal, or base-60 system, the Babylonians were able to solve various types of functions and equations through substitution and Diophantine method.

By looking at how the Babylonians created their own system of equations, it makes me wonder if math is all about generalization and abstraction. I think we have to consider both aspects depending on the context in which they are used. In such a way, we can use generalization for finding a pattern of addition and write out an equation for it. We can also use abstraction to explain hundreds of different scenarios under the same structure in math. However, since modern math is involved with other things such as proofs and measurements, it is safe to say that math is about a lot of things, not just generalization and abstraction.

When we're discussing general and abstract relationships, algebra is often involved. However, I believe that using words and statements is one way to express general and abstract relationships. It is also possible to use verbal and visual presentations to prove the relationships, such as using a diagram to represent the area of a triangle. Although these relationships can be proved without algebra, I think doing so becomes more complicated when solving problems.

As we watch our students struggle and learn at the same time, sometimes we teachers question ourselves whether students are learning in the way we want them to. I suppose students may go through similar stages of rhetorical and syncopated algebra as they move toward learning and doing symbolic algebra. Through the use of concrete numerical examples, students can gradually connect their understanding of the symbolic algebra, which plays an important role in modern math involving symbol expressions.

Some history on time calculation

When thinking about time itself, I have always had the thought of it being a continuous and endless cycle in the back of my mind. In relation to my own experience, I have always understood a year as a cycle consisting of 365 days, with 12 months and 4 seasons: spring, summer, fall and winter. Each season lasts 3 months, which is equivalent to one quarter of the cycle, and each month has roughly 30 days. In addition, I think of one month as 1/12 of a cycle. I also think of one hour as 60 minutes, which represents one cycle of the clock, one minute as 60 seconds which is 1/60 of the cycle and one second as one tiny division of the analog clock.

While reading the articles, both articles mentioned Babylonians refining the sexagesimal (base 60) system inherited from the Sumerians. However, MacTutor focused more on the history of the Babylonians and how they developed the sexagesimal system, while Scientific American focused on the development of time through the course of history. For instance, in MacTutor's article, he mentioned suggestions that base 60 was chosen because people thought there were 360 days a year back in the day, while the other article mentioned how 60 was useful in different ways. Essentially, MacTutor provided a possible reason for implementing base 60 through different theories and explanations, while Scientific American emphasized on the usefulness of 60.

One thing that I find really interesting is how Egyptians used a T-shaped bar to divide the interval between sunrise and sunset into 12 parts, calculating the duration of daylight. At night, they marked the remaining 12 stars of the set of 36 stars, resulting in 12 divisions of night, which eventually led to the creation of the 24 hour period. I have always thought the 12 hour time interval was created from the movement of the Sun from sunrise to sunset. The way the ancient civilization measured time is not something I would've ever thought of, considering that technology wasn't as advanced as it is today. This made me realize the way I think about time actually comes from systems that people developed by observing the Sun and stars. When I think about how the day and night pass, I will definitely make the connection and picture the stars at night.

Babylonian Multiplication Table

 

Crest of Peacock Response

While I was reading this article, the first thing that surprised me was how mathematical activities have mostly been ignored and devalued outside of Europe. The article stated how Europeans have neglected the contributions from Arabia, India and China while they viewed Western society as the universal model for power. I was particularly shocked because when I was growing up, I was always taught the importance of Arabic numbers and how they influenced modern math. On top of that, the article mentioned that there have been recent studies showing early Indian mathematics already existed around 800 to 500 BC, which was around the same period as the earliest known Greek mathematic. Without giving enough credit to its origins, it is illogical to claim mathematic itself as an exclusively Western product. It is important for us to understand the true history, background and origin of math.

To add on to what I wrote in the first paragraph, I find the contributions from India and Arabia to math history very intriguing. I was surprised to see how much the Arab world contributed to bringing in different techniques of equations to the modern world, which included the adoption and development of algebra, as well as concepts including the "Algebraic number" that we know of today. On the other hand, India's involvement with the origin of math also caught my eye. It was stated in the article that numerical concepts and methods, such as Arabic numerals, actually originated from India and not Arabia. This took me by surprise, because I have always thought Arabic numbers originated from Arabia. I also searched up the geographical location of India and surprisingly, as mentioned in the article, it was an important meeting place between different regions, which made the influence of transmission easier. 

This brings me to the last thing that caught my attention: the amount of transmission of knowledge between different nations. While we all now understand that India and Arabia made significant contributions to the Western society, I was surprised when the article also mentioned the transmission between Egypt and Mesopotamia because I was barely taught about the history between the two nations. Although there wasn't any direct evidence of mathematical exchange between the two, the spread of Mesopotamian algebra and geometry to Egypt indicated it was possible that they had made connections with each other. In Figure 1.4 of the article, it was clear that there was transmission between Greece, Egypt and Mesopotamia as well. To add on to that, the figure also revealed the cross-cultural interaction between China and India, including Chinese translations of Indian work on math. Given how much interactions there were between nations, I am now not surprised to see math developing and spreading through the world. The transmission of knowledge between each nation in the early days created an interesting period in mathematical history that has impacted our lives to this day.

Why Teach Math History Response

Prior to reading the article about integrating math history into the classroom, my thoughts were that math history should be integrated into the teaching of math, as I believed that it provides us with a deeper understanding of the concepts we are learning and the origins of where they come from. Not only that, by incorporating it into my own teaching, it also gives students the opportunity to learn about the historical aspects which can potentially make learning more interesting. As for how we can incorporate math history into our own teaching, one way I can do this is by allowing students to acknowledge the historical context of math. Beyond that, I believe the rest will come along as I start teaching.

While reading, the one thing that struck me was where the article mentioned how there had been different arguments made on incorporating history while teaching math. Based on some of the arguments, we knew that reasons include how there was a lack of time and history and math are not connected to each other. However, it made me wonder if there are ways to incorporate into math classes somehow, such as shortening lecture time and spend ten to fifteen minutes each class discussing interesting facts about the history of math. On the other hand, the part mentioning historical snippets and how textbooks have incorporated them also interested me, especially when discussing about the style and design of the snippet. While integrating history into math is beneficial, we have to look at how the contents are portrayed to students, to which I agree with the article. A lot of the time, the portrait can often decide if the materials are worth learning, an easily appealing context can easily catch students' attention when reading the textbook, which in turn makes them more interested in learning. 

After reading the article, I now believe that incorporating historical content into math is crucial, as understanding the origin of some of the theories and equations are important for learning. At the same time, revisiting old concepts also allow students to discover the mistakes from the concepts and discuss why they were wrong. These are beneficial as students would become encouraged in analyzing and problem-solving at the same time.