Chapter 40: Geometric Progression

The twelve female guards were stunned for a moment.

The name "Golden Hairpins" suggested by His Highness was indeed much more pleasant than "Sword Hairpins." Although the word "sword" was dropped, the sword itself belonged to "metal" among the Five Elements, so the meaning was not lost. Moreover, "gold" possessed not only the rigid courage of blades and spears but also the overbearing authority of royalty and the prosperous aura of abundant wealth. As the closest guards to the most noble princely mansion in Great Xu, who wouldn't want to be courageous, authoritative, and wealthy all at once?

Without any objection, the twelve heroines nodded in succession.

Zhuque could see that the girls were exceptionally fond of the title "Twelve Golden Hairpins," to the point of being impatient.

"You lot, hurry up and thank His Highness for bestowing such a name upon you!"

*Swish! Swish! Swish!*

The twelve heroines immediately placed one fist against their chests and knelt in unison.

"We thank Your Highness for the grace of bestowing the name 'Twelve Golden Hairpins'!"

*They bend their bows to fight like men, dreaming of the days they once drew their brows.*

From this moment on, the Twelve Golden Hairpins were a team.

Chai Xiao gestured for them to rise and laughed. "Why do you all seem so unexcited about this first big money-making venture? Do you feel that after such a grand display, earning only 170 taels from ticket sales is disappointing? It's not even as much as the 300 taels you took out for your wager earlier, is it?"

The Twelve Golden Hairpins looked at each other, wanting to nod but not daring to. His Highness had indeed voiced what they were thinking. If the earnings were less than the investment, wasn't that a loss?

Chai Xiao smiled slightly, pinching the large leaf between two fingers and waving it rhythmically. "Why don't you try thinking a bit further? What if this is only the revenue from the first day, the very beginning of the round-robin tournament?"

The first day? If they earned this much at the very start, and there were many days to follow, the total would certainly be considerable!

The eyes of the twelve Golden Hairpins widened, as if scavengers who had been struggling in the wilderness for a long time had suddenly discovered a mountain of gold.

The large leaf in Chai Xiao's hand began its first rotation.

"The reason the admission fee is low initially is to attract more people to watch and sign up! Outside the three paid seating areas, any commoners who wish to watch can do so for free, standing or sitting as they please, provided they don't crowd or collide and keep the entrances to the paid zones clear. I believe that at first, people will just come to watch the spectacle. But as they realize it gets more and more exciting, the paid areas will quickly fill up. As the crowd grows, more people will be eager to sign up for the stage."

"The first day of the round-robin tournament will consist of encounter matches. Since the combat strength of the participants will vary greatly, the outcomes will be decided quickly. To ensure both safety and entertainment, and to prevent contestants from hiding and avoiding combat, we can set two rules. First, stop at the point of contact! Second, decide the winner within twenty moves! The weapons used will all be wooden—knives, swords, spears, and staves—dipped in lime before the match to determine the winner based on points marked on the body. Alternatively, being knocked off the stage counts as a loss, or being knocked down three times results in a defeat. This way, we aim to finish the round-robin matches on the first day."

"If there are enough registrants, we can set the subsequent elimination rounds for the top thirty-two or even the top sixty-four."

The large leaf in Chai Xiao's hand began its second rotation.

"Take the round of thirty-two to sixteen as an example! On the second day, we hold sixteen matches! The rules of stopping at the point of contact, being knocked off the stage, and being knocked down three times remain unchanged. However, the twenty-move limit will be extended to fifty, and it will be a best-of-three round system to increase the spectacle! Most importantly, all ticket prices will double! That means the third-tier seats will cost 20 wen, second-tier parlor seats 100 wen, and the VIP section will start at 200 wen. This is equivalent to 200 wen for the fifth row, 400 for the fourth, 600 for the third, 800 for the second, and 1,000 wen—or one tael of silver—for the first row. Can you calculate how much the ticket revenue will be on this second day?"

Jian Yu quickly provided the answer. "While it seems like everything is increasing, calculating each item step-by-step is troublesome. But since everything is doubling, we don't need to calculate each item; just double the total income from the first day to get the final sum! Twice 170 taels is 340 taels."

Chai Xiao applauded loudly. "Clap! Clap! Clap! Excellent! Simplify the complex! Precision doesn't have to be complicated; as long as you grasp the method, you can simply double it!"

Following this, Chai Xiao stopped his hand and began the third rotation with the leaf.

"The third day, sixteen to eight! All ticket prices double again! Third-tier seats, 40 wen! Second-tier parlor seats, 200 wen! VIP section starting at 400 wen! This is equivalent to 400 wen for the fifth row, 800 for the fourth, 1,200 for the third, 1,600 for the second, and 2,000 wen—or two taels of silver—for the first row. Calculate again, what is the revenue for the third day?"

Jian Yu and several others who had already grasped the pattern answered directly: "Twice 340 taels is 680 taels."

Chai Xiao held the leaf and began the fourth rotation. "The fourth day, eight to four. Everything doubles again. Calculate the revenue!"

At this point, Jian Yu led almost all the Golden Hairpins in responding: "Twice 680 taels is 1,360 taels."

Chai Xiao began the fifth rotation. "The fifth day, four to two."

Before his voice had even faded, the Twelve Golden Hairpins had learned to "snatch the answer." "Twice 1,360 taels is 2,720 taels!"

Chai Xiao made the final, sixth rotation. "The sixth day, the final championship!"

The Twelve Golden Hairpins answered in unison: "Twice 2,720 taels is 5,440 taels."

The large leaf in Chai Xiao's hand stopped moving. "Now, add up the income from these six days. What is the total?"

All the Golden Hairpins were stunned and looked toward Jian Yu.

What is easy for those who know is difficult for those who don't. Jian Yu quickly laid out the truth. "The first day is 170 taels, the second day is twice that, the third is four times that... the sixth day is thirty-two times 170 taels, which is 5,440 taels. The base of 170 remains constant, so we only need to sum the six numbers: one, two, four, eight, sixteen, and thirty-two. The sum of the first five is thirty-one, just one less than thirty-two. Therefore, we only need to calculate twice the income of the sixth day, which is 10,880 taels, and then subtract 170 taels, resulting in 10,710 taels."

The other Golden Hairpins listened to this calculation with wide-eyed amazement, nodding repeatedly.

Chai Xiao merely smiled. This wasn't difficult. It was simply the summation of a geometric series where the first term is one and the common ratio is two. It was equivalent to two to the power of zero, plus two to the power of one, and so on, up to two to the power of six. Following the formula: one minus two to the power of six, divided by one minus two, multiplied by the first term. It equaled two to the power of seven minus one, which is sixty-three. If there were a seventh day, the revenue would double to 10,880 taels, and subtracting the first day's 170 taels would yield the answer Jian Yu provided: 10,710 taels.

Jian Yu didn't know about power series or geometric ratios. She simply relied on the basic distributive law of multiplication and addition to realize that the sum of the first five days was just one unit less than the sixth day's revenue. So, one only needed to double the sixth day's income and subtract the first day's income. It was simple, effective, and the fastest arithmetic method.

Chai Xiao suddenly felt a strange sensation, as if time had converged. He felt as though he had returned to the days of his summer break, volunteering to teach in the mountains for two months. Back then, he was teaching junior high students some intriguing techniques and methods of high school mathematics. The lesson on the blackboard had been exactly this: the general term and summation formula for geometric series.

Time quickly snapped back to the current Great Qi. He began the most crucial "enhancement" part of his "lecture" to the Twelve Golden Hairpins.

The large leaf in his hand began its seventh rotation!