Magus Tech (Technological Wizard)

The two-minute game went on like this for a long time, until most of the sun in the west had sunk, leaving only a red edge shining, which stretched the shadows of the chess pieces on the chessboard.

"Snapped!"

Li Cha made a move again, the clown saw the problem when he swept the board, and said bluntly: "I lost again, but it doesn't matter, let's do it again! I don't believe it, I can't win a game." As he spoke, the clown started Clean up the chessboard and pick up all the white pieces.

However, Li Cha refused aloud: "No need for Mr. Clown, even if I play a hundred more games, as long as I go first with black stones, you will definitely lose. This has nothing to do with chess skills, but has something to do with mathematics. It has been calculated."

"Huh?" The clown stopped picking up white stones, looked at Li Cha, and asked, "What do you mean?"

"What I mean is that although the rules of backgammon are simple, there are loopholes in the calculations. Without any restrictions, black stones who go first will win, yes, they will win-as long as they follow a specific way, and any Everyone is sure to win.

Generally speaking, for the sake of fairness, in a normal game, sunspots will have 'three-three bans', 'four-four bans', 'four-three-three bans', 'four-four-three bans', and 'long consecutive bans' .

The so-called "three-three-forbidden-hand" means that when a black piece lands a piece, it will form two or more three-linked pieces at the same time, which is absolutely impossible for the white piece to resist under the premise of the first move.

Therefore, in a normal game, once Heizi walks out of the "three-three forbidden hand" or is forced to walk out of the "three-three forbidden hand", it will be judged as a loser. But now in the game between us, if there is no forbidden hand, then you naturally have no hope of winning. "

"Can this be calculated?" The clown still didn't quite believe it.

"In fact, the simpler the game, the easier it is to calculate." Li Cha spread his hands, "If you have any doubts, we can play another game."

"What?"

"A game of chess pieces." Li Cha said, he took the chessboard away as he spoke, grabbed a handful of chess pieces, counted a total of nineteen pieces, and divided them into three piles.

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

"The rules are like this." Richard said, "The two of us take turns to take chess pieces from these three piles, and each time we can choose any pile to take. The minimum number of chess pieces to take is one, and the maximum number of piles is taken away. Whoever takes the last chess piece left will lose."

After a pause, Li Cha looked at the clown and continued: "This game is different from backgammon,

But it can also be calculated through mathematics, so the first move is sure to win. "

The clown's eyes flickered, and he stared at the chess pieces for a while and thought for a while: "I don't believe it, I have thought of several ways for the second player to win right now."

"In this case, let's give it a try." Li Cha said, and took five of the nine pieces from the pile on the far right first. The three piles of chess pieces immediately turned into a "three seven five" look.

The clown thought for a while and took away the remaining four chess pieces on the far right, turning into a "37".

Li Cha then took away the four chess pieces in the second pile, turning into a "three three" appearance.

The clown stared at the chess piece for a long time, but did not make a move. He looked at Li Cha and asked, "Am I considered a loser now?"

"Yes." Li Cha nodded, "Although there is no result yet, you have already lost.

Now it's a 3:3 situation. If you take away one pile at a time, I can take away two of the remaining pile and leave one for you, and you lose. If you take two of one pile and I take the other pile and leave one for you, you lose.

Out of caution, if you only take one piece from one pile, I will also take one piece from the other pile, making it 2:2.

And 2:2 and 3:3, there is no essential difference-you take one pile, I take one from the other pile, leave one for you, and you lose. You take one of the piles, I take all the other pile and leave one for you, and you still lose. No solution. "

"But it's just one situation, there are other situations. It's impossible to win just because you go first." The clown insisted.

Li Cha shook his head: "It's true that this is not the only situation, all situations can be roughly divided into three types, that is, the winning form left behind at the end of evolution.

The first form, the most extreme, leaves you with a pile, and only one chess piece. Whether you take it or not, you lose.

In the second form, you are left with two piles, as long as the number of the two piles is the same, no matter how many there are, they will all become 2:2 in the end, or in the first form, you lose.

The third form, the remaining three piles, is more complicated, but in the end it will gradually become the first two. As long as the first move does not make mistakes, the second mover will still lose. "

"But how can you not make a mistake when you go first?" The clown's voice became louder, and he lost his composure, "You don't know what I think, how can you control my holding method? And since you can't control my holding method, so what? Can you guarantee that you will not make mistakes every time you take it?"

"It's not so complicated." Richard said, "In the calculation, I don't need to know how you take it, I just need to give you an equilibrium state. To put it simply, first, I convert the number of each pile of chess pieces into binary Arranged horizontally. By the way, binary is to change the "every decimal into one" into "every binary into one", for example."

As Li Cha said, he used a slender chess piece to write three lines of numbers on the ground:

0011 (3)

0111 (7)

1001 (9)

Then draw a horizontal line under the three rows of numbers, and add the numbers under the horizontal line to become a number:

1123

"There is such a definition that a number that is a multiple of 2 is an even number, and if it is not, it is an odd number. Among the four numbers of 1123, if there is an odd number, it is a non-equilibrium state.

And what I want to do is to bring you into a balanced state by taking chess pieces. For example, I took out five pieces from the third pile at the beginning, and it became a balanced state, that's it. "

While talking, Li Cha erased the written numbers, and wrote them again with slender chess pieces:

0011 (3)

0111 (7)

0100 (4)

——

0222

"Now all the numbers are even numbers, which is a balanced state. In the balanced state, no matter how you take it, what I do is to continue to change it back to the balanced state. In this way, in the end, you will lose.

So, it doesn't matter how you take it, as long as I go first and come up with a balanced state, you will lose. This is why the first move wins - it's not me that beats you, it's math, it's binary.

This is simpler than the principle of backgammon, and the amount of calculation is less, you should be able to figure it out. "

After Li Cha finished speaking, the clown was silent.

The clown, who was silent, stared at the numbers written by Li Cha for a long time, then took the chess pieces and started writing on the ground, and then separated three piles of chess pieces to simulate continuously. The result of the simulation is the same as what Richard said - every time the first move is sure to win.

In the end, the clown looked at Li Cha with a complicated expression.

Li Cha said: "Mr. Clown, you see, it's hard for me to beat you when I play 'clown chess' with you, but if you play games like 'backgammon' and 'pick pieces' with me, you will always Can't win. 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 you played. And I also know that I will win when I play "backgammon" and "take pieces" with you. This is caused by the difference in areas of expertise, and it can also be regarded as a different home field. That's why I don't quite agree with what you said at the beginning. "

"By the way." Li Cha looked at the clown and asked seriously, "By the way, Mr. Clown, you still remember what you said to me, right?"

"I—" The clown stood up, looked at Li Cha, his eyes changed several times in a row, finally took a deep breath, and suddenly turned around, "The wooden stage is built, it's time for me to perform, the game is over first, yes Chance... we will talk next time."

"Is there a next time?" Li Cha asked.

"Maybe." The clown said, walking towards the wooden platform without looking back.

At this time, the sun in the west had completely set, and the sky went black. Li Cha looked at the clown's back and narrowed his eyes slightly.

... ()