Summary

Many people think mathematics is difficult, but mathematics can be solved purely through logical thinking. Most people who dislike math actually dislike this logical way of thinking. However, if you avoid things just because they’re difficult, you won’t be able to accomplish anything. Let’s explore how to think logically through examples.

Cutting tofu problem

Let’s consider the following problem: If you want to cut a cube-shaped block of tofu into 27 smaller cubes of equal size and shape, what is the minimum number of cuts required? With a little thought, you can quickly see that the answer is six. But if you ask why it’s six, this problem turns into more than just a simple puzzle.
The small cube at the center is a cube, so it has six faces. Each of these faces was created by cutting and was not part of the original block. Since there are six faces, six cuts are required.

A tale of a farmer, the daughter to the farmer, and a wealthy man

There was a farmer who struggled to make a living after his crops failed due to a prolonged drought. To survive, he borrowed money from a wealthy man in the village and managed to get through the year. However, the following year, the same disaster struck, and he once again went to the rich man to borrow money. This time, the wealthy man refused, saying the farmer had not yet repaid the loan from the previous year.

Desperate, the farmer pleaded with him, and the rich man proposed a wager: if the farmer won, all his debts would be cleared, and he would receive another loan for the year. But if he lost, the rich man would take the farmer’s beautiful daughter as his servant. With no other choice, the farmer agreed.

The wager was as follows:

• a white stone and a black stone would be placed into a pouch, and the farmer’s daughter would draw one. If she picked the white stone, the farmer would win, and the loan would be granted. But if she picked the black stone, not only would the farmer have to repay his debt immediately, but his daughter would also become the rich man’s servant.

To ensure his victory, the wealthy man secretly placed two black stones in the pouch. However, the wager ended in the farmer’s daughter’s favor. How was this possible?

The clever daughter said,

“I’m afraid to put my hand inside the pouch. I’ll choose a stone from the outside instead.”

She selected a stone without looking and then turned the pouch inside out. Naturally, a black stone fell out. She then said,

“Since the stone that fell out of the pouch is black, the one I picked must have been white.”

Unable to refute her logic, the wealthy man had no choice but to forgo the debt.

Why we learn mathematics

Mathematics is simple and clear, just like this. However, due to the rote-learning education we have received, we tend to solve all problems using memorized mathematical formulas. This is why people find math difficult. To excel in mathematics, one must develop understanding through logical thinking rather than relying on memorization.

To cultivate logical thinking in mathematics, a ‘thread of thought’ is essential. One of the many reasons we study mathematics is to develop the ability to find and follow this thread.

Another reason for studying mathematics is to develop the ability to view the world rationally. Everything in the world follows unchanging principles, and we study mathematics to acquire the ability to explain these principles in a reasonable and logical manner. While other fields such as natural sciences, humanities, and social sciences can also achieve this, mathematics holds a superior level of precision. As mentioned earlier, mathematics is the same for everyone, whereas other disciplines involve a degree of subjectivity. Without a proper understanding of these principles, humanity would not have progressed.

To think mathematically means following a logical process to find the solution to a problem. Let’s explore this concept through an example.

The tale of Gauss in elementary school and the symmetry of arithmetic sequence

The German mathematician Carl Friedrich Gauss, often called the “Prince of Mathematics”, is considered one of the three greatest mathematicians in history, alongside Archimedes and Newton. This is a story from when he was just ten years old.

Gauss’s teacher, who wanted to slack off, assigned the students the task of summing the numbers from 1 to 100. While the other students slowly added 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10, and continued in this tedious manner for a long time, Gauss immediately figured out that the answer was 5050 and effortlessly solved the problem. Seeing this, the teacher recognized Gauss’s genius.

How was this possible?

Gauss discovered a pattern in the problem. He noticed that 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on. He realized that this pattern continued throughout the sequence. Since there were 50 pairs of 101, he simply calculated 101 × 50 = 5050 to obtain the answer instantly. He had used the symmetry of arithmetic sequences, which is also the key idea behind deriving the formula for the sum of an arithmetic series.

Gauss thought mathematically, logically, and rationally. We previously mentioned that mathematics requires a thread of thought. The reason we study mathematics is to develop the wisdom to discover these interconnected threads.

Conclusion

Anybody can think of the matters in the real world mathematically and find the threads of thought. Mathematics, borne out of need, helps us organize our thoughts into an organized struture.

Therefore, we should not merely study mathematics as a subject to be learned, but rather develop the ability to observe and interpret phenomena mathematically, enabling us to solve real-world problems in a rational and logical manner.

Implication

Through learning mathematics, we can learn how to think logically.

Underlying context – Background behind

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Notions – Key ideas

Thread of thought: Learning the rest from learning individual concepts

Index – Source(s)

How to Think Like Mathematicians / 피타고라스 생각 수업
09 How to Think Logically Like Gauss | Thread of Thought / 가우스처럼 논리적으로 생각하는 법 | 생각의 끈

Trajectory – Where I’m headed now

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