Chapter 217 The question of the bridge

Zhang Fangping looked at the master beside him, and the master also had an unbelievable expression, so he turned back again: "You talk about it first." ”

Su You said: "This kind of question, we call it residual theory. The simple and easy-to-understand solution is as follows: first list the numbers divided by three and the remainder two: two, five, eight, eleven, ......"

"Then list the numbers divided by five and the remainder three: three, eight, thirteen, eighteen......"

"Of these two columns, the common number that appears first - eight."

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

"List the strings as: eight, twenty-three, thirty-eight, ......"

"Then list the numbers two, nine, sixteen, twenty-three, and thirty divided by seven remainder ......two."

"This results in the minimum number of commons that meets the conditions of the question – twenty-three."

"Of course, this is a stupid solution, there is actually another solution to this problem, there is a song recipe to explain: three people walk together seventy rare, five plum blossoms twenty-one branches, seven characters reunion moon is half, in addition to one hundred and five to know."

"In the first sentence, three people go together seventy, which means that the number is divided by three, and the remainder is multiplied by seventy."

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

"The third sentence, the reunion of the seven sons is half of the moon, is to divide the number by seven, and the remainder is multiplied by fifteen."

"In the fourth sentence, divide by one hundred and five, then add up the above three products and subtract the multiple of one hundred and five, and the difference is the number sought."

"If I use the arithmetic column formula of the land temple......

After speaking, he took out a notebook and pencil from his schoolbag, and wrote an equation with a brush: "Oh, that's it." ”

The master took the notebook and saw that it was written: 2×70+3×21+2×15=233,233-105×2=23.

The master was able to understand this magical formula, and asked carefully: "Dare to ask Childe, seventy, twenty-one, fifteen, where do these numbers come from?" Why multiply it by two, three, and two? Why subtract one hundred and five after that? ”

Su You said with a smile: "Seventy divided by three is more than one, which can be divided by five and seven; Therefore, twice the number of seventy, which can be divided by three and remainder two, is also divisible by five and seven, and the condition of the first remainder is satisfied, without considering the last two remainders;

In the same way, twenty-one is divided by five and is divisible by three and seven; So the three times of twenty-one can be divided by five and the remainder of three, and at the same time it can be divisible by three and seven; This satisfies the second remainder condition, regardless of the first and third remainders; ”

"Fifteen divided by seven is divisible by three and five at the same time, so that twice as many as fifteen can be divided by seven and two and is divisible by three and five at the same time; This satisfies the third remainder condition, regardless of the first and second remainder conditions. ”

"The first three verses of the poem illustrate this situation separately, and then add them together, which satisfies both the first divisible part of the question and the last three remainder conditions."

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

Su You said with a smile: "This number is already the answer, but it is not the smallest answer, so it is necessary to subtract the common multiple of the three numbers, that is, one hundred and five or its multiples, and subtract it to the point where it cannot be subtracted, which is the minimum answer, which is the meaning of the last poem." ”

The master was so excited that he danced with his hands: "This is the truth!" That's the truth! In the past, the patchwork method could only solve one problem, and if the number was too large, it would be time-consuming and laborious. With this method today, all kinds of problems can be solved! Wonderful! It's just whimsy! ”

After speaking, he looked at Su You with a blank eye: "Childe, just now you said that this question is a category...... Surely you know a lot of these questions, right? ”

Su You said: "It can be seen that Mr. is also a studious person, so I'll write you a few." ”

After speaking, I wrote a few words in the notebook.

Today, there are unknown things, the remaining two of five or five numbers, the remaining two of seven or seven numbers, and the remaining four of nine or nine numbers.

Han Xin ordered soldiers, more than two people in a group of three, more than three people in a group of five, and more than four people in a group of seven.

There are unknown things today, the remaining two of the three or three numbers, the remaining one of the four or four numbers, and the remaining number of the twelve?

The master's mental arithmetic ability is quite powerful, he grabbed Su You's pencil and read the questions while columnist, and he solved the first two questions, shouting happily.

When I looked at the third question, I was dumbfounded again: "Uh, Childe, this third question is different from the previous questions......"

Su You took the pen and chuckled: "Actually, it's still the same type, but there are some small changes, this is called an extended question type - I'll show you...... Huh, got it? In fact, it is still not far from its roots, and it is difficult to understand the Fa, this kind of question is difficult for people. ”

The master waved again and again: "Thank you Son, thank you Childe, it's really a god!" ”

Su You said with a smile: "I'm a big Song person who is good at math, that's the amount of the car, I'm 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 a layer of window paper, a little bit of penetration. ”

The master's face was full of flattery: "Gongzi's words are too modest, this is an exam question for the imperial court's Ming arithmetic subject, and most of the candidates in the Great Song Dynasty take the exam with written skills, relying on rote memorization of answers." ”

"The old man has heard that there are a class of smart people in my Great Song Dynasty, who can solve a class of people in one way, and they are all geniuses, but I didn't expect to see them in person today, which is really overjoying."

Zhang Fangping put his hand on his forehead and said to Su Xun with a smile: "I don't know who is taking the test ...... whom."

The master turned his head and smiled: "Where does the little boy still need to be tested, when I am a master, I deserve it, my question is simply ridiculous and generous......"

Zhang Fangping laughed and said: "Hugh has to be long, and others will destroy their own prestige!" Go and bring another question. ”

The master said "ah": "Which way?" ”

Zhang Fangping squeezed his eyes: "That stone is estimated." ”

The master said: "Ming Gong, that's ......"

Zhang Fangping glared: "Go!" ”

The master hurriedly responded, and after a while, he came in with a scroll of drawings: "This, please take a look at the little boy." ”

Su You opened the drawing, and above it was an arched bridge.

The master said: "Childe, you see, this is an arch bridge, nine zhang across the river, the highest part of the bridge is two zhang from the water surface, the bridge is one zhang and five zhang wide, you 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 swelled when he heard this, and he felt that it was impossible not to go to the bridge to measure it himself: "Mingrun!" Hugh is babbling! ”

Su You took out the compass and ruler from his schoolbag and drew a picture in his notebook: "Let's not look at the width of the bridge first, can we simplify this question like this?" Know the chord length of the arc, know the arch height, find the arc length of the arc? ”

The picture was simple and straightforward, and everyone around nodded their heads.

Su You smiled: "This requires knowing a few theorems, first of all, any point on the circle, connected with the two ends of the diameter, its angle is a right angle, we can prove as follows." ”

After speaking, he explained the Dharma to everyone.

Zhang Fangping was still a little skeptical, so he drew a few more circles with a compass, and then connected them with a pencil and a ruler: "That's really the case." ”

Su You said with a smile: "Ming Gong, this science is called geometry, and the language used is called logic, which is similar to the theory of hard whiteness. ”

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

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

Everyone nodded.

"Another example, a finite line segment can be extended, right?"

The crowd nodded again.

"There are three more."

"Draw a circle with any point as the center and any distance;"

"All right angles are equal to one another;"

"A straight line intersects two other straight lines in the same plane, and if the sum of the two interior angles on one side is less than the sum of two right angles, the two straight lines intersect on this side after an infinite extension."

Everyone thinks these things are simple, and they don't know why this baby wants to mention them.

Su You said: "With these five axioms, we can derive countless theorems. A theorem is a proposition or formula that can be proved to be correct through deduction and deduction through axiomatic logical limitations. ”

"For example, just now we proved the right angle on the circle, then it becomes a theorem, and the theorem is also a true proposition, so no matter how Zhang Gong draws, under the conditions I gave, only right angles can be drawn."

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

Su You arched his hand and said: "Zhang Gongming sees that there is only one possibility to move it. ”

After saying a finger to the five axioms written on the paper: "Unless they are wrong!" ”