The two-minute game continued like this for a long time, until most of the sun in the west sank down the mountain, leaving only a red edge shining, making the shadows of the chess pieces on the chessboard grow long.
"Snapped!"
Richard made another move, and the clown saw the problem as soon as he glanced at the chessboard, 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 began Clean up the chessboard and pick up all the white stones.
But Richard refused out loud: "No need, Mr. Clown, even if there are another hundred games, as long as I move first with the black stones, you will definitely lose. This has nothing to do with chess skills, but with mathematics. It is calculated."
"Huh?" The clown stopped picking up the white pieces, looked at Richard, and asked aloud, "What do you mean?"
"What I mean is that although the rules of backgammon are simple, there are loopholes in the calculation. Without any restrictions, the black stone who moves first is bound to win. Yes, it is bound to win - as long as you play in a specific way, you can win with anyone. No matter what, you will win.
Generally speaking, for the sake of fairness, in a normal game, black players will have 'three-three forbidden hands', 'four-four forbidden hands', 'four-three-three forbidden hands', 'four-four-three forbidden hands', and 'long-continuous forbidden hands' .
The so-called 'Three Three Forbidden Hands' means that when Black drops a piece, two or more triples will be formed at the same time. This is something that White cannot resist if he moves first.
Therefore, in a normal game, once the black stone breaks out of the 'three-three forbidden moves' or is forced to move out of the 'three-three forbidden moves', it will be judged as a loss. But in the current match between us, there is no hand ban, so naturally you have no hope of winning. "
"Can this be calculated?" The clown still didn't believe it.
"In fact, the simpler the game, the easier it is to calculate." Richard spread his hands and said, "If you have any doubts, we can play another game."
"What?"
"A game with chess pieces." Richard said, taking the chess board away as he spoke, grabbing a handful of chess pieces, counting a total of nineteen pieces, and dividing them into three piles.
The first pile has three chess pieces, the second pile has seven chess pieces, and the third pile has nine chess pieces.
"The rules are like this." Richard said, "The two of us take turns to take chess pieces from these three piles. Each time, we can choose any pile to take. The minimum number of chess pieces taken is one, and the maximum number is to take all the pieces from one pile. Whoever gets the last piece left will lose."
After a pause, Richard looked at the clown and continued: "This game is different from backgammon, but it can also be calculated mathematically, so the one who plays first is bound to win."
The clown's eyes flashed and he stared at the chess pieces and thought for a long time: "I don't believe it. I have thought of several ways to win with a second hand."
"In that case, let's give it a try." Richard said, taking five of the nine chess pieces from the rightmost pile first. The three piles of chess pieces immediately looked like "three, seven, five."
The clown thought for a while and took away the four remaining chess pieces on the right, turning them into "three-seven" shapes.
Richard then took away the four chess pieces from the second pile and turned them into "three three".
The clown stared at the chess pieces for a long time without making a move. He looked at Richard and asked, "Am I considered a loser now?"
"Yes." Richard nodded, "Although the results haven't come out yet, you have already lost.
Now it is a 3:3 situation. If you take away a bunch at once, I can take away two of the remaining piles and leave one for you. You lose. If you take two of the pile, I take the other pile and leave one for you, and you lose.
Out of caution, if you only take one chess piece from one pile, then I will take one chess piece from the other pile, making it 2:2.
There is no essential difference between 2:2 and 3:3 - you take one pile, I take one from the other pile, leaving one for you, and you lose. You take one of the piles, and I take all of the other piles, leaving one for you, and you still lose. No solution. "
"But this is just one situation. There are other situations. It's impossible to win just because you go first." The clown insisted.
Richard shook his head: "Indeed, this is not the only situation. All situations can be roughly divided into three types, that is, the winning form left by the first mover at the end of the evolution.
The first form, the most extreme, leaves you with a bunch of chess pieces 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 two piles have the same number, no matter how many there are, it will all become 2:2 in the end, or in the first form, you lose.
The third form, with three piles left, is a little more complicated, but it will eventually turn into the first two. As long as you don't make mistakes in the first move, you will still lose in the second move. "
"But how can you not make mistakes when you go first?" The clown's voice became louder, slightly rude, "You don't know what I think, how can you control my way of holding it? And since you can't control my way of holding it, so what? Can you guarantee that you won’t make mistakes every time you get it?”
"It's not that complicated." Richard said, "In the calculation, I don't need to know how you get it, I just need to give you a balanced 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 ordinary calculation's 'add one to every ten' to 'add one to every two', like this."
As Richard spoke, 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, add the numbers under the horizontal line, and it becomes a number:
1123
"There is such a definition. If a number is a multiple of 2, it is an even number. If not, it is an odd number. Among the four numbers 1123, if there is an odd number, it is a non-equilibrium state.
What I have to do is to create a balanced state for you by taking chess pieces. For example, at the beginning I took out five from the third pile and it became a balanced state, and that's it. "
As he spoke, Richard erased the numbers he had written and wrote them again with a slender chess piece:
0011(3)
0111(7)
0100(4)
—
0222
"Now all the numbers are even numbers, which is a state of equilibrium. In the state of equilibrium, no matter how you take it, what I do is to continue to change it back to the equilibrium state, so that in the end, you will lose.
So, it doesn’t matter how you take it, as long as I take the first move and come up with a balanced state, you will lose. This is the reason why the first player will win - it is not me who defeats you, but mathematics and binary.
This is simpler than the principle of backgammon and requires less calculation. You should be able to figure it out. "
After Richard finished speaking, the clown fell silent.
The clown in silence stared at the numbers written by Richard for a long time, then he took the chess pieces and started writing on the ground, and then divided out three piles of chess pieces to simulate continuously. The result of the simulation is the same as what Richard said - the first mover will win every time.
Finally, the clown looked at Richard with a complicated expression.
Richard spoke: "Mr. Clown, as you can see, it's hard for me to outplay you when I play 'Clown Chess' with you, but if you play games like 'Gobang' and 'Get Pieces' with me, you will never be able to defeat me." You 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 even if I play 'Gobang' or 'Get the pieces' with you. This is caused by differences in areas of expertise, and can also be seen as differences in home fields. That's why I don't really agree with what you said at the beginning. "
"That's right." Richard looked at the clown and asked seriously, "By the way, Mr. clown, you still remember what you told me, right?"
"I -" the clown stood up, looked at Richard, his eyes changed several times, and finally took a deep breath and suddenly turned around, "The wooden platform is built, it's time for me to perform, let's end the game first, if there is a chance … We’ll talk next time.”
"Is there a next time?" Richard asked.
"Maybe." The clown said, walking towards the wooden platform without looking back.
At this time, the sun in the west completely set, and the sky turned dark. Richard looked at the clown's back and narrowed his eyes slightly.
…