Jing Ke lifted his eyes to gaze at the gigantic suspended stone that filled his field of vision. Against the moving clouds in the sky, the dark mass appeared oppressive.
Jing Ke turned to the king and said, “Your Majesty has spared my life once, and even if I could finish the calculations in time, it wouldn’t be enough to erase the crime of my attempt on your life. I’m not afraid to die. Please give me five more days to think about this. If I still can’t come up with a plan, I’ll sit under the monument willingly.”
Four days later, Jing Ke requested an audience with the king, which was immediately granted. Calculating the circular ratio was the most important task on the king’s mind.
“From your expression, I gather that you have indeed come up with a plan.” The king smiled.
Jing Ke did not answer directly. “Your Majesty, you once said that you would give me all the resources that I require. Is that still the case?”
“Of course.”
“I need three million men from your army.”
The number did not astonish the king. His eyebrows only lifted briefly. “What kind of soldiers?”
“The common soldiers under your command now are sufficient.”
“I think you should be aware that most of the soldiers in my army are illiterate. In two years, you cannot possibly teach them complex mathematics, let alone finish the calculations.”
“Sire, the skills they would need can be taught to the least intelligent soldier in an hour. Please give me three soldiers so that I can demonstrate.”
“Three? Only three? I can easily give you three thousand.”
“I only need three.”
King Zheng waved his hand and summoned three soldiers. They were all very young. Like other Qin soldiers, they moved like order-obeying machines.
“I don’t know your names,” Jing Ke said, tapping the shoulders of two of the soldiers. “The two of you will be responsible for number input, so I’ll call you ‘Input One’ and ‘Input Two.’” He pointed to the last soldier. “You will be responsible for number output, so I’ll call you ‘Output.’” He shoved the soldiers to where he wanted them to stand. “Like this. Form a triangle. Output is the apex. Input One and Input Two form the base.”
“You could have just told them to stand in the Wedge Attack Formation,” King Zheng said, glancing at Jing Ke with a smile.
Jing Ke took out six small flags—three white, three black—and handed them out to the three soldiers so that each held a black flag and a white flag. “White represents zero; black represents one. Good. Now listen to me. Output, you turn around and look at Input One and Input Two. If they both raise black flags, you raise the black flag as well. Under all other circumstances, you raise the white flag.”
Jing Ke repeated the instructions one more time to be sure the three soldiers understood. Then he began to shout orders. “Let’s begin! Input One and Input Two, you can raise whichever flag you want. Good. Raise! Good. Raise again! Raise!”
Input One and Input Two raised their flags three times. The first time they were black-black, the second time white-black, and the third time black-white. Output reacted correctly each time, raising the black flag once and the white one twice.
“Very good. Sire, your soldiers are very smart.”
“Even an idiot would be capable of that. Tell me, what are they really doing?” King Zheng looked baffled.
“The three soldiers form a component for a calculation system, which I call an AND gate. If both numbers input into the gate are one, the result output is also one; otherwise, if one of the numbers input is zero, such as zero-one, one-zero, or zero-zero, the result output is zero.” Jing Ke paused to let King Zheng digest the information.
The king said impassively, “All right. Continue.”
Jing Ke turned to the three soldiers again. “Let’s form another component. You, Output, if you see either Input One or Input Two raise a black flag, you raise the black flag. There are three situations where that will be true: black-black, white-black, black-white. When it’s white-white, you raise the white flag. Understand? Good lad, you’re really clever. You’re the key to the correct functioning of the gate. Work hard, and you will be rewarded. Let’s begin operation. Raise! Good, raise again! Raise again! Perfect. Your Majesty, this component is called an OR gate. Whenever one of the two inputs is one, the output is also one.”
Then Jing Ke used the three soldiers to form what he called a NAND gate, a NOR gate, an XOR-gate, an XNOR-gate, and a tristate gate. Finally, using only two soldiers, he made the simplest gate, a NOT gate: Output always raised the flag that was opposite in color from the one raised by Input.
Jing Ke bowed to the Emperor. “All calculating components have been demonstrated. This is the scope of the skills that the three million soldiers must learn.”
“How can you perform complex calculations using such simple, childish tricks?” King Zheng’s face was full of distrust.
“Great King, the complexity in everything in the universe is built up from the simplest components. Similarly, an immense number of simple components, when structured together appropriately, can generate extremely complex capabilities. Three million soldiers can form a million of these gates I’ve just demonstrated, and these gates can be put together into a whole formation capable of any complex calculation. I call my invention a calculating formation.”
“I still don’t understand how the calculations will be carried out.”
“The precise process is complicated. If Your Majesty continues to be interested, I can explain in detail later. For now, it’s enough to know that the operation of the calculating formation is based on a novel method of thinking about and writing down numbers. In this method, only two numerals, zero and one, corresponding to the white and black flags, are needed. But this new method can use zero and one to represent any number, and this allows the calculating formation to use a large number of simple components to collectively carry out high-speed calculations.”
“Three million is almost the entirety of my arm, but I will give them to you.” King Zheng sighed meaningfully. “Hurry. I’m feeling old.”
A year passed.
It was another beautiful day with the sun and the moon both out. King Zheng and Jing Ke stood together on a high stone dais, with the king’s numerous ministers in rows behind them. Below them, a magnificent phalanx of three million Qin soldiers stood arrayed on the ground, the entire formation a square three miles[11] on each side. Lit by the freshly risen sun, the phalanx remained still like a giant carpet made of three million terra-cotta warriors. But when a flock of birds wandered above the phalanx, the birds immediately felt the potential for death from below and scattered anxiously.
Jing Ke said, “Your Majesty, your army is truly matchless. In an extremely short time, we have completed such complex training.”
King Zheng held on to the hilt of his long sword. “Even though the whole is complex, what each soldier must do is very simple. Compared to the military training they went through, this is nothing.”
“Then, Your Majesty, please give the great order!” Jing Ke’s voice trembled with excitement.
King Zheng nodded. A guard ran over, grabbed the hilt of the king’s sword, and stepped backward. The bronze sword was so long that it was impossible for the king to pull it out of the scabbard without help. The guard knelt and presented the sword to the king, who lifted the sword to the sky and shouted, “Calculating formation!”
Battle drums began to beat, and four giant bronze cauldrons at the corners of the stone dais came to life simultaneously with roaring flames. A group of soldiers standing at the edge of the dais facing the phalanx chanted in unison, “Calculating formation!”
11
For the benefit of English readers, Chinese measurement units (such as