How to break into a math book

After Tu Hua explained the meaning of the poem, Wang Boyu quickly figured out the answer: "It's very simple if you explain it clearly!"

“可以用方程来解答！”王博宇分析道,“假设最顶端那层有x盏灯，按照题目给出的条件，它下面那层的灯数是它的二倍,也就是第二层有2x盏灯；第三层的灯数又是第二层的二倍,就是2×2x盏灯，即4x盏；依次类推，下面的每一层分别有8x、16x、32x、64x盏灯。”

“整座塔总共有7层，所有灯的数目加起来是381，也就是说x2x4x8x16x32x64x=381。很简单，x的值为3,也就是说最顶端那层塔上有三盏灯！”

Wang Boyu calculated the answer excitedly, and looked down at the white five-pointed star under his feet with some uncertainty: "So 3 is the number represented by this white astrolabe?"

As soon as he finished speaking, Tang Bo, who was standing on the yellow stone platform, also found that the five-pointed star under his feet moved.An identical brocade box rose from the stone platform, and there was still a piece of white paper in the box, on which it was written: "Today, there is a woman drinking a cup on the river. A Jin official passed by. The woman said: There are guests at home. The Jin official said: How many guests?" The woman said: Two people share a meal, three people share a soup, and if you use five cups, I don’t know how many guests are there?”

After Tang Bo finished reading the words on the paper, he translated it casually: "A woman was washing dishes by the river, and a Jin official passed by. The woman said that she had guests at home, and the Jin official asked how many guests were there? Woman He replied, two people share a bowl for eating, three people share a bowl for soup, a total of 5 bowls are used, how many guests do you think there are?"

“1/2x1/3x=5。”唐博随口就说出了方程式，想都没想，“总共有6个客人。”

"So this yellow stone platform under my feet represents the number 6?"

These math problems from ancient classics are actually very simple, but Tu Hua feels that the real crux of this level does not lie in this.They have to get the numbers represented by the seven stone platforms according to the prompts. If they don't get the prompts, they will be completely unknown about the numbers on the stone platforms.

Therefore, it is far more important to judge the appearance of each number on this circular chessboard than to guess the number by hints, because according to the current rhythm, they will definitely not be able to get all the hints.

At this time, the light blue stone platform under Shen Siyi's feet also moved.The astrolabe opened, revealing the same wooden brocade box as the previous two.Shen Siyi opened the paper in the box, and when he saw the words on it, his brows frowned: "Today, there are [-] paces from Tianguang, and [-] paces from (sound longitudinal zong), what is Tian Geo?"

Whether this question is translation or algorithm, there is no need to consider it. It means that there is a field with a width of 15 steps and a length of 16 steps. How big is the field.

Very simple multiplication, 15×16=240, that is to say, the area of ​​this field has 240 steps.But the crux of the problem lies in the base system of measurement units in ancient times. We usually understand the units of measurement as centimeters, decimeters, and meters, but the distance units in ancient times are different from those of today.

古代对于距离单位的表达通常是用“尺”、“寸”“丈”来表示，过去的1尺折合成现在的距离单位来说，约等于23.1厘米。10寸等于1尺，10尺等于1丈。而这里所说的“步”也是一种丈量单位，1步约等于6尺。

Converted to the unit of farmland, 240 square steps equals 1 mu.

The point where meditation is easy to get tangled is what is the unit of this question, is it a step?ruler?Zhang or mu?Because the question stem does not give the exact unit of expression, the numbers obtained by different units are different.This will affect the value of the light blue stone platform under Shen Siyi's feet.

Sun Wei thought for a while, and said: "If you strictly follow the information in the question, I think the unit of this answer should be 'square step'. The information in the question is all steps, and the answer should correspond to it."

Tu Hua shook his head: "I don't think so. If I remember correctly, this question should come from "Nine Chapters of Arithmetic". 240 square steps are a relatively large and complicated number for the ancients. In real life People don’t measure it like this. Besides, 240 square feet equals 1 mu, and I think people at that time should use 1 mu to express it.”

Shen Siyi also agreed with Tu Hua's statement: "The ancients did not know how to use decimals and fractions, so they could only change the unit to the next level when calculating. For the expression of farmland area, 'mu' is indeed universal. .”

"So..." Shen Siyi looked down, "The light blue stone platform under my feet should be the number 1?"

All the prompts for the second round have ended.Among the seven astrolabes, they can currently determine three values. The first is the white stone platform at the center of the circle at Wang Boyu’s feet, and the white represents the number 3; the second is the yellow stone platform at Tang Bo’s feet, and the yellow color represents 6; the last is The light blue stone platform under Shen Siyi's feet, light blue represents 1.

These three stone platforms are not in a straight line, and there are too few conditions, and they are currently unable to judge the regularity between the numbers.Everyone remained in place, and soon the third round of dice rolls began.

After the third round of throwing, they found that luck was still on their side, and the five people were on two corresponding straight lines again.This time, Shen Siyi is in the white position in the center of the circle. Wang Boyu is standing on the yellow stone platform, and Sun Wei is in the light yellow position. The three of them are standing on the straight line of yellow, white, and light yellow.

This time, Tu Hua stood on the red stone platform, and Tang Bo stood on the pink stone platform. The two of them and Shen Siyi standing in the center formed a straight line of red, white, and pink.

Among these two straight lines, the yellow line of Wang Boyu and Sun Wei has already appeared in the previous round, and the yellow and white positions have given hints, leaving only the light yellow five-pointed star at Sun Wei's feet still blank of.

Sure enough, a brocade box emerged from the stone platform at Sun Wei's feet.There is only a simple sentence written on the paper: "Today, there are four feet for the strands and five feet for the strings. What is the number of sentences?"

This is the earliest explanation of the Pythagorean Theorem in "Nine Chapters of Arithmetic".The "strand" in this sentence represents the longer side of the right triangle, and the "chord" represents the hypotenuse of the right triangle.

而“句”也就是我们在勾股定理中所说的“勾”，是直角三角形中较短的那条直角边。所以这道题目的答案显而易见，勾股定理最常见的勾股数就是3、4、5，即3的平方4的平方=5的平方。

How many "sentences" are asked in the title, it should be the number 3.

The numbers under Sun Wei's feet were quickly determined. In this way, the three points on the yellow, white, and light yellow lines were all determined. The numbers represented by these three points were 6, 3, and 3 respectively.

With only such a set of numbers, it is difficult to distinguish the rules.Tu Hua secretly expected that in their straight line of red, white, and pink, the stone platform under his and Tang Bo's feet would be able to pop up a reminder.

Disappointment always comes after making a wish. In this straight line, only a brocade box popped out from the red stone platform under Tu Hua's feet.The title in the brocade box is similar to Sun Wei's title: "Today there is a sentence with three feet and a share with four feet. What is the string?"

Therefore, the number represented by the red stone platform at the foot of Tuhua is 5.

At this point, the game of three rounds of dice is over. For the seven astrolabe numbers in the seven-star array, they only got 5, which are yellow, white, and light yellow that form a straight line, representing the numbers 6, 3, and 3. 5; The adjacent two colors are red and light blue, representing numbers 1 and [-].

The pink stone platform corresponding to the red and the blue stone platform corresponding to the light blue have no answer.

Xu Guangqi on the side laughed and said, "Have you got the answer to the seven-star array?"

站在红色位置上的涂化凝眉：“按照从左至右的顺序，圆周上的数字分别是6、5、1、3，圆心处的数字也是3。这些数字中有什么规律吗？”

Tang Bo thought for a while and said, "The order in which we jump the grid is white, red, yellow, blue, pink, light yellow, and light blue. Could it be that these numbers are arranged in such a sequence?"

“3、5、6、蓝、粉、3、1这样一串数字，是不是要我们在这串数字中寻找规律，来推断出蓝色和粉色位置的数？”

Wang Boyu said in frustration: "But judging from the numbers we know so far, there is no law in this sequence."

Sun Wei rarely agrees with Wang Boyu: "I also think it should not be the arrangement of numbers."

"Think about the actual situation of this level. The person who invited us here is Xu Guangqi, a scholar who studies circular trigonometric functions." Sun Wei glanced at Xu Guangqi, "Combined with this strange circular formation under our feet, I I feel that the laws of these numbers must be related to circles."

Tu Hua looked up at her: "How do you say it?"

Sun Wei laughed and said, "Look at the three-point straight line where I, Shen Siyi and Wang Boyu are standing. What is it to this circle?"

"diameter?!"

"Yes." Sun Wei nodded, "The three straight lines appearing in this circle are the diameters of this circular platform. Therefore, the numbers represented by the diameters should be equal. The sum of the three numbers on the branch line we are on is 633=12, This means that the sum of the numbers on the other two diameters should also equal 12."

"Blue, white, light blue, blue unknown, white is number 3, light blue is number 1, then blue should be equal to 8. Red, white, pink, red is number 5, white is equal to 3, then pink Should represent the number 4."

Sun Wei's analysis seemed reasonable, but Tu Hua always felt that something seemed to be missing.If the law is judged according to the diameter theorem of a circle, what do the colors of the round platforms under their feet represent?

Why are there three colors of red, yellow, and blue, and why is there a color system formed between each straight line?Color must be a clue that cannot be ignored, but Ke Hua still can't figure out what the connection is.

Just when everyone was about to agree with Sun Wei's statement, Shen Siyi suddenly interrupted everyone's train of thought: "I think...the circle is just a blindfold."