The two-minute game went on like this for a long time, until most of the sun sank in the west, and only a red edge shone brightly, stretching the shadows of the chess pieces on the chessboard.

"Smack!"

Richard fell again, and the clown swept the chessboard, saw the problem, 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 single game." As he spoke, the clown began to pack up the chessboard and pick up all the whites.

Richard refused: "No need Mr. Clown, even if you play another hundred games, as long as I go first with the sunspot, you will definitely lose, this has nothing to do with chess, it has to do with mathematics, it is calculated." ”

"Huh?" the clown stopped picking up the white child, looked at Richard, and asked aloud, "What do you mean?"

"What I mean by this is that the rules of backgammon are simple, but there are holes in the calculations, and without any restrictions, the first black disc is guaranteed to win, yes, it must win - as long as it is played in a certain way, with anyone, it is a sure 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-even bans'.

The so-called 'three-three forbidden hands' means that when a sunspot falls a son, two or more triplets will be formed at the same time, which is under the premise of taking the lead, and the white son cannot resist at all.

Therefore, in a normal game, once a sunspot 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 loss. But now that we're playing against each other, there's no banning, so you don't have any hope of winning. ”

"Can that be calculated?" the clown was still a little unconvinced.

"In fact, the simpler the game, the easier it is to calculate. Richard said, "If you're in doubt, we can play another game." ”

"What?"

"A game with chess pieces. Richard said, taking the chessboard away as he spoke, grabbed a handful of chess pieces, counted nineteen in all, and divided them into three piles.

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

"The rules are like this. Richard said, "The two of us take turns taking pieces from these three piles, and each time we can choose any one of them." The number of pieces to be taken is at least one, and the maximum number of pieces is taken by a pile. Whoever takes the last remaining piece loses. ”

After a pause, Richard looked at the clown and continued: "This game is not the same as backgammon, but it can also be calculated mathematically, so the first move is also a sure win." ”

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

"In that case, let's give it a try. Richard said, taking five of the nine pieces on the far right from a pile. Three piles of chess pieces immediately turned into the appearance of "three-seven-five".

The clown thought about it and took the remaining four chess pieces on the far right, turning into the appearance of "Sanqi".

Richard then took the four pieces from the second pile, turning into a "three-three" look.

The clown stared at the chess pieces for a long time, did not make a move, looked at Richard and asked, "Am I losing now?"

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

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

Out of an abundance of caution, if you take only one piece from one pile, then I will take one piece from another pile, making it 2:2.

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

"But it's just one situation, there are other situations, and you can't win just because you make the first move. The Joker insisted.

Richard shook his head: "It's true that this is not the only situation, the whole situation can be roughly divided into three types, that is, the winning form left by the first hand in the end."

The first form, the most extreme, gives you a bunch of leftovers and only one piece. Whether you take it or not, you lose.

The second form gives you the remaining two piles, as long as the two piles are the same number, no matter how many, in the end they all become 2:2, or the first form, you lose.

The third form, the remaining three piles, is a little more complicated, but in the end it will become the first two little by little, as long as the first hand does not make a mistake, you will still lose the second hand. ”

"But how can you not make a mistake in the first move?" the clown's voice became louder and slightly out of shape, "and you don't know what I think, how can you control my handling?

"It's not that complicated. "In the calculations, I don't need to know how you take it, I just give you an equilibrium." To put it simply, first, I convert the number of pieces in each pile into a binary horizontal arrangement. By the way, binary is to turn the ordinary calculation of 'every ten into one', like this. ”

Richard said, using 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 lines of numbers, and add the numbers under the horizontal line to become a number:

1123

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

And what I have to do is to conjure up a balanced state for you by taking the chess pieces, for example, at the beginning I took five from the third pile and turned it into a balanced state, and that's it. ”

As he spoke, Richard erased the numbers and wrote again with a slender chess piece:

0011（3）

0111（7）

0100（4）

——

0222

"Now all the numbers are even, it's a kind of equilibrium, and in the case of equilibrium, no matter how you take it, what I do is continue to change it back to equilibrium, so that in the end, it's you who loses.

So, it doesn't matter how you take it, as long as I take the lead and come up with a balance, you'll lose. That's why the first move wins - it's not me who beats you, it's math, it's binary.

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

Richard finished speaking, and the clown was silent.

The clown in silence stared at the numbers written by Richard for a long time, and then took the chess pieces himself and began to write on the ground, and then divided into three piles of chess pieces and continued to simulate. The result of the simulation, as Richard said, was the first to win every time.

Eventually, the Joker looked at Richard with a complicated expression.

Richard spoke: "Mr. Clown, you see, it's hard for me to play 'clown chess' with you, but if you play 'backgammon' and 'take chess pieces' with me, you will never win." The reason for this has nothing to do with wisdom and humility, it is only a matter of choice.

The reason why you chose to play 'clown chess' with me is because you knew you were going to win before you played. And I also know that I will win by playing 'backgammon' and 'taking chess pieces' with you. This is due to the difference in the field of expertise, and it can also be seen as the difference in the home field. That's why I don't 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 remember what you told me, right?"

"I-" The clown stood up and looked at Richard, his gaze changed several times in a row, and finally he took a deep breath and turned around suddenly, "The wooden platform is built, I should perform, let's end the game first, there is a chance...... We'll talk next time. ”

"Is there a next time?" asked Richard.

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

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

......