Exploring Technology in a Wizard World
Chapter 486 - 485: It’s Not Me Who Beat You, It’s Binary (Math-oriented, Subscribe with Caution, Skip if Disliked)

Chapter 486: Chapter 485: It’s Not Me Who Beat You, It’s Binary (Math-oriented, Subscribe with Caution, Skip if Disliked)

The match every two minutes continued for a very long time, until most of the sun in the west had sunk behind the mountains, leaving only a glowing red rim, stretching the shadows of the chess pieces across the board.

"Snap!"

Richard placed his piece once again, and Clown swept the board with his gaze, immediately spotting the problem, and declared, "I’ve lost again, but it doesn’t matter, let’s go again! I refuse to believe that I can’t win a single game." With those words, Clown began to tidy up the board, picking up all the white pieces.

However, Richard voiced his refusal, "No need, Mr. Clown. Even if we play another hundred games, as long as I play the black pieces first, your loss is certain. This has nothing to do with chess skill; it’s about mathematics, a calculated certainty."

"Hmm?" Clown stopped picking up the white pieces and looked at Richard, asking, "What do you mean?"

"What I mean is, although the rules of Five-in-a-row are simple, they have calculated flaws. Without any restrictions, the first player, holding the black pieces, will inevitably win—yes, win inevitably—as long as they play in a specific manner, against anyone.

Generally speaking, for fairness, in standard matches the black pieces are subject to ’Triple Three Forbidden Hand,’ ’Four-four Forbidden Hand,’ ’Four-Three-Three Forbidden Hand,’ ’Four-four-three Forbidden Hand,’ and ’Long Connection Forbidden Hand.’

The so-called ’Triple Three Forbidden Hand’ refers to the situation where the black pieces, upon playing a single piece, simultaneously form two or more open rows of three, which is utterly indefensible for the white pieces under the first move advantage.

Therefore, in a standard match, once the black pieces play a ’Triple Three Forbidden Hand,’ or are forced to play a ’Triple Three Forbidden Hand,’ they are deemed to have lost. But since there are no forbidden moves in our matches, you naturally have no hope of winning."

"This can be calculated?" Clown still seemed a bit incredulous.

"As a matter of fact, the simpler the game, the easier it is to calculate," said Richard, spreading his hands. "If you have doubts, we could switch to another game."

"What game?"

"A nim game," said Richard, and as he spoke, he removed the board and grabbed a handful of pieces, counting a total of nineteen, dividing them into three piles.

The first pile had three pieces, the second had seven pieces, and the third had nine pieces.

"The rules are as follows," Richard explained, "The two of us take turns taking pieces from these three piles. You may choose any pile each time. The number of pieces taken must be at least one, up to the entire pile. Whoever takes the last piece loses."

Pausing, Richard looked at Clown and continued, "This game, unlike Five-in-a-row, can also be calculated mathematically, so the first player is guaranteed to win."

Clown’s eyes twinkled as he stared at the pieces, pondering for a while: "I don’t believe it. I’ve already thought of several strategies for the second player to win."

"In that case, let’s try it," Richard said. He started by taking five pieces from the pile on the far right of the nine pieces. The three piles of pieces immediately changed to a "three-seven-five" formation.

Clown thought for a moment, then took the remaining four pieces from the far right, changing the piles to "three-seven."

Richard followed by taking four pieces from the second pile, turning it into a "three-three" situation.

Clown stared at the pieces for a long while, not making a move. He looked up at Richard and asked, "Does that mean I’ve lost already?"

"Yes," Richard nodded. "Although the outcome hasn’t been played out yet, you’ve already lost.

Now it’s a 3:3 situation. If you take away all of one pile at once, I can take the remaining two from the other pile, leaving you one, resulting in your loss. If you take two from one pile, I take the other pile, leaving you one, and you lose.

Out of caution, you take only one piece from a pile. Then I’ll take just one piece from the other pile, making it 2:2.

And 2:2 and 3:3 are essentially the same—If you take an entire pile, I’ll take one from the other pile, leaving you one, and you lose. If you take one from a pile, I’ll take the rest, leaving you one, and you still lose. No solution."

"But that’s just one scenario. There should be others. It can’t be that just because you go first, you’re guaranteed to win," Clown insisted.

Richard shook his head, "Indeed, it’s not just this one scenario. All the scenarios can be largely divided into three types, which are the winning formations left by the first player as the game evolves.

The first formation is the most extreme, leaving you with one pile with a single piece. Regardless of whether you take it or not, you lose.

The second formation leaves you with two piles, and as long as the number of pieces in both is equal, regardless of the quantity, it all eventually becomes 2:2, or the first formation. You lose.

The third formation is with three piles remaining, a bit more complex, but ultimately, it still erodes down to the first two types. As long as the first player doesn’t make a mistake, you, the second player, will still lose."

"But how could you avoid mistakes with the first move?" Clown’s voice grew louder, showing a slight loss of composure. "You don’t know what I’m thinking, how can you control my play? And since you can’t control how I play, how can you guarantee that you won’t make any mistakes every time?"

"It’s not that complicated," Richard said. "In the calculation, I don’t need to know how you play, I just need to give you a state of balance. Simply put, first, I convert the quantity of each pile of pieces into a binary horizontal arrangement. Right, binary is where you change the regular counting system of ’carry one every ten’ to ’carry one every two,’ like this."

As he spoke, Richard used a slim piece to write three lines of numbers on the ground:

(3)

(7)

(9)

Then he drew a horizontal line below the three lines of numbers and added the numbers below the line, which turned into:

"There’s this definition, a number that is a multiple of 2 is even, and if not, it’s odd. Among the four digits of 1123, there are odd numbers, and that is a non-balanced state.

And what I need to do is to take pieces and produce a state of balance for you, for example, if I start by taking five from the third pile to reach balance, like this."

As he spoke, Richard erased the numbers he’d written, and used the slim piece to write again:

(3)

(7)

(4)

——

"Now all the numbers are even, which is a state of balance. In a state of balance, no matter how you play, all I need to do is keep returning it to a state of balance. In the end, you’ll be the one to lose.

So, it doesn’t matter how you play, as long as I go first and bring about a state of balance, you’ll lose. That’s the reason why the first move guarantees victory—it’s not me who’s beating you, but mathematics, binary.

This is simpler than the principle of Five-in-a-row, and it requires less calculation. You should be able to understand it."

Having finished, Richard looked at Clown, who was silent.

In the silence, Clown stared at the numbers Richard had written for a long time, then took pieces and began to write on the ground, dividing them into three piles and continually simulating. The results of the simulation were the same as what Richard had said—every time, the first move guaranteed victory.

Eventually, Clown looked at Richard with a complex expression.

Richard spoke up, "Mr. Clown, you see, it’s hard for me to beat you at ’Clown Chess,’ but if you play games like ’Five-in-a-row’ or ’taking pieces’ with me, you’d never win either. The reason for this has nothing to do with wisdom or humility; it’s just a matter of choice.

The reason why you chose to play ’Clown Chess’ with me is because you knew you would win before playing. And I likewise know that if I play ’Five-in-a-row’ or ’taking pieces’ with you, I will also win. This is due to the difference in fields of expertise and can be seen as having a different home field advantage. That’s why I don’t quite agree with what you said at the beginning."

"Right." Richard looked at Clown seriously and asked, "By the way, Mr. Clown, you still remember what you originally told me, right?"

"I—" Clown stood up, looked at Richard, his gaze changing several times, then took a deep breath and suddenly turned around, "The stage is set, it’s time for my performance. Let’s end the game for now, we can talk another time if we have the chance."

"Will there be another time?" Richard asked.

"Perhaps," Clown said, without looking back as he walked toward the stage.

At that moment, the sun in the west had completely set, and the sky darkened along with it. Richard watched Clown’s retreating figure, squinting his eyes.

...