Chapter 217: Bridge Problem

Zhang Fangping looked at the master next to him, who also had a bewildered expression, and then turned back: "Tell me first."

Su You said: "We call this type of question the remainder theory. The simple and easy-to-understand solution is as follows: first list the numbers that have a remainder of two when divided by three: two, five, eight, eleven..."

"List the numbers that leave three when divided by five: three, eight, thirteen, eighteen..."

"In these two columns of numbers, the common number that appears first is eight."

"The least common multiple of three and five is fifteen. When the two conditions are combined into one, it is an integer multiple of fifteen, plus eight."

"This list of numbers is: eight, twenty-three, thirty-eight..."

"Then list the numbers two, nine, sixteen, twenty-three, thirty..."

"This yields the smallest public number twenty-three that meets the conditions of the question."

"Of course this is a stupid explanation. There is actually another way to solve this problem. There is a rhyme that explains: Three people walking together are seventy-seven, five trees have twenty-one plum blossoms, seven characters are reunited, the moon is half, divide it by one hundred and five and you will know.

"

"The first sentence, three people walking together is seventy rare, which means divide the number by three and multiply the remainder by seventy."

"The second sentence, five plum blossoms and twenty-one branches, is to divide the number by five, and multiply the remainder by twenty-one."

"The third sentence, when the seven sons reunite on the first half of the month, divide the number by seven, and multiply the remainder by fifteen."

"In the fourth sentence, if you divide by one hundred and five, you can get it. Then add the above three products and subtract the multiple of one hundred and five. The difference obtained is the number you are looking for."

"If we use the calculation formula of Tutu Temple..."

After saying that, he took out a notebook and pencil from his schoolbag and scribbled down a calculation: "Here, that's it."

The master took the book and saw it said: 2x70+3x21+2x15=233,233-105x2=23.

The master was actually able to understand this magical formula, and he raised his hand and asked carefully: "May I ask, sir, where do these numbers seventy, twenty-one, and fifteen come from? Why are they divided into two, three, and multiplied by two? Later, because of the

Why subtract one hundred and five?"

Su You said with a smile: "Seventy divided by three leaves one, which is divisible by five and seven. Therefore, twice seventy, which can be divided by three and leaves two, is also divisible by five and seven, which satisfies the first remainder condition.

, without considering the last two remainders;

"Similarly, twenty-one divided by five leaves one, which is divisible by three and seven at the same time. Therefore, three times twenty-one can be divided by five and leaves three, and is also divisible by three and seven. This satisfies the requirement

Two remainder conditions, without considering the first and third remainders;"

"Fifteen divided by seven leaves a remainder of one, which is also divisible by three and five. Therefore, twice fifteen, divided by seven, leaves a remainder of two and is divisible by three and five at the same time. This satisfies the third remainder condition, and

There is no need to consider the first and second remainder conditions."

"The first three lines of the poem explain this situation respectively, and then adding them together, this not only satisfies the divisibility part of the problem at the beginning, but also satisfies the three remainder conditions at the end."

The master suddenly realized: "Wonderful! This idea is amazing!"

Su You smiled and said: "This number is already the answer, but it is not the minimum answer. Therefore, we have to subtract the common multiples of three numbers, that is, one hundred and five or its multiples. When it is reduced to the point where it cannot be reduced, the minimum answer is

, this is the meaning of the last line of the poem."

The master danced with excitement: "This is the ultimate truth! This is the ultimate truth! The previous patchwork method can only solve one problem. If the number is too large, it will be time-consuming and laborious. Now that we have this method, we can solve all kinds of problems."

All the problems can be solved! It’s wonderful! It’s simply a wonderful idea!”

After saying that, he looked at Su You eagerly: "Sir, you just said that this question is of the same type... You must know many more questions of this type, right?"

Su You said: "It can be seen that sir is also a studious person, so I will write a few words for you."

After I finished speaking, I wrote a few words in my notebook.

Now there is an unknown object with an unknown number. There are two remaining in the number of five and five, two in the number of seven and seven, and four in the number of nine and nine. What is the geometry of the object?

Han Xin ordered his troops. There were two men in a group of three, three men in a group of five, and four men in a group of seven. How many troops were there?

Now there are things with unknown numbers. The number of three or three is two, and the number of four or four is one. How many are the twelve numbers?

The master's mental arithmetic ability is very good. He grabbed Su You's pencil and looked at the problems while listing the equations. He solved the first two problems in a swipe and screamed with joy.

When I saw the third question, I was dumbfounded again: "Well, sir, this third question is different from the previous questions..."

Su You finished writing and chuckled: "Actually, it's still the same type, with just a few minor changes. This is called expanding the question type. Let me explain it to you... Well, got it? In fact, it's still the same.

If you know how to understand it, this kind of question will not stump anyone."

The master bowed repeatedly: "Thank you so much, sir, thank you so much, sir, it's really a stroke of luck!"

Su You said with a smile: "I, the Song Dynasty people who are good at mathematics, are just like me. I am nothing. It's just that mathematics is difficult to spread, so you don't know it. In fact, for people with a foundation in mathematics,

This is just a layer of window paper, it will show through just a little bit."

The master's face was full of flattery: "Young Master's words are too modest. This is an examination question of the imperial arithmetic department, and most of the candidates in the Song Dynasty took the exam with their writing skills and relied on rote memorization of answers to pass the test."

"I have heard that there is a smart person in the Song Dynasty who can use one method to interpret another type. He is a genius. Unexpectedly, I can see him in person today. I am really overjoyed."

Zhang Fangping put his hand on his forehead and said to Su Xun, dumbfounded: "I don't even know who is testing whom..."

The master turned his head and smiled and said: "Young master, you still need to take the test, you are worthy of being my master. My questions are simply making people laugh..."

Zhang Fangping said with a smile: "Don't let other people's ambitions destroy your own prestige! Go and get another question."

The master said "Ah": "Which way?"

Zhang Fangping winked: "That's the stone estimate."

The master said: "Ming Gong, that is..."

Zhang Fangping glared: "Go quickly!"

The master hurriedly agreed, and after a while he came in with a roll of drawings: "This, young master, please take a look."

Su You opened the drawing and found an arch bridge on it.

The master said: "Sir, look, this is an arch bridge, spanning ninety feet across the river. The highest point of the bridge is two feet away from the water, and the bridge is one and a half feet wide. We need to calculate how many stones are needed to lay the bridge deck."

Su You said: "This is much simpler than the one just now."

Su Xun's head felt swollen after hearing this, and he felt that it would be impossible if he didn't go to the bridge to measure it himself: "Mingrun! Don't talk nonsense!"

Su You took out a compass and a ruler from his schoolbag and drew a picture on his notebook: "Ignoring the width of the bridge, can we simplify this question into this? Knowing the chord length of the arc and the arch height, what is the answer?

The arc length of the arc?”

This picture is simple and clear, and everyone who gathered around nodded.

Su You smiled and said: "This requires knowing a few theorems. First, if any point on the circle is connected to the two ends of the diameter, the angle formed by it is a right angle. We can prove it as follows."

After speaking, explain the proof method to everyone.

Zhang Fangping was still a little doubtful, so he drew a few more circles with a compass, and then connected them with a pencil and ruler: "That's true."

Su You smiled and said: "Ming Gong, this science is called geometry, and the language used is called logic, which is similar to Jianbai's theory."

"For plane geometry, there are only a few basic true propositions, which we in the Earth Temple call axioms."

After speaking, write on the paper: "For example: you can draw a straight line from any point to any other point. Right?"

Everyone nodded.

"For another example, a finite line segment can continue to be extended, right?"

Everyone nodded again.

"There are three more."

"A circle can be drawn with any point as the center and any distance;"

“All right angles are equal to each other;”

"If a straight line intersects two other straight lines in the same plane, and the sum of the two interior angles on a certain side is less than the sum of the two right angles, then the two straight lines will intersect on that side after being infinitely extended."

Everyone thought that these things couldn't be simpler, and they didn't know why this kid would mention these things.

Su You said: "With these five axioms, we can deduce countless theorems. Theorems are propositions or formulas that can be proven to be correct through deduction and deduction through axiomatic logic restrictions."

"For example, if we just proved a right angle on a circle, it becomes a theorem, and the theorem is also a true proposition. Therefore, no matter how Mr. Zhang draws it, under the conditions I gave, he can only draw a right angle."

Zhang Fangping is also a very smart person: "No wonder countless people in ancient and modern times are obsessed with mathematics. This is the pursuit of eternal principles!"

Su You cupped his hands and said, "Zhang Gongming saw that there is only one possibility to move it."
Chapter completed!