【英譯】

Mengxi Bitan Chapter 18 Jiyi 7

According to a story, the Buddhist monk Yi Xing [1] of the Tang dynasty calculated ?the total number of different configurations that may be placed on a Go ?board. I have thought of the problem as well, and found the method of ?calculation rather straightforward. However, the number of configurations is ?so large that it cannot be properly expressed by the numbers that are ?commonly known.

Here I shall present some examples. If the board is a 2x2 square grid, ? then it has four positions, which can produce 81 different configurations. If ?the board is a 3x3 square grid, then it has nine positions, which can produce ?19,683 different configurations. If the board is a 4x4 square grid, then it ?has sixteen positions, which can produce 43,046,721 different configurations. ?If the board is a 5x5 square grid, then it has twenty-five positions, which ?can produce 847,288,609,443 different configurations. (According to the ancient number system: yi is 10 multiplied by 10,000, zhao is 10 multiplied by 100,000,000, ti [2] is 10,000 multiplied by zhao. ?However, for mathematicians, yi is ?10000 multiplied by 10,000, zhao is ?10,000 multiplied by 10,000 yi, and ?gai is 10,000 multiplied by 10,000 zhao. Here we use the calculation ?method of the mathematicians.) If the board is 6x6 ?square grid, then it has thirty-six positions, which can produce 150,094,635,282,031,926 [3] ?different configurations. If the board is a 7x7 square grid or larger, the ?number of different configurations is too large to be written out. Exhausting ?the 361 positions of a true Go board, the number of configurations is ?approximately the number 10,000 being written consecutively forty-three times[4].

(This so-called “forty-three 10,000s” ?means the last 10,000 is 10,000, the second is , the third is , the fourth is , the fifth is , ?the sixth is , which is called one hai, the seventh is , the eighth is , ?the ninth is . Beyond that there is no name to express it; however, the product of 10000 to the power of 43 is the approximate order of magnitude of this large? number while the remaining digits are not included.)

The method: first consider a point, which can have three possible ?configurations. (Black, ?white, or empty.) Each additional position, regardless if it is ?horizontal or vertical, will contribute an additional multiple of 3. When ?this is increased to 361 points, the total number of configurations is obtained ?by a multiple of 3 for every point.

Another method: First calculate the number of configurations of one ?side and take it as the “base number” (法 fa). (There are 19 points in one line, so the number of configurations is 1,162,261,467.) ?If one more line is added to it, multiply it by the “base number”. ?When it is multiplied 19 times, we get the same number.

Another method: First multiply the “base number” by itself. (The product is 1,350,851,717,448,287,334 [5], which is the number of ?configurations for two lines with a total of 38 positions.) Then ?use the product as the multiplier and multiply the “base number” by the ?multiplier, and get the second result. After that, multiply the second ?product by itself and get the third product. Next use the third product as the ?multiplier and multiply it with the second product to get the fourth product. ?The total number of configurations is obtained after multiplying the product ?by itself and then multiplying it with the “base number”. Among these ?methods, this one is the quickest. (Only five calculations were made ?to arrive at the answer for 361 points.) Of all the myriad ?changes, the total number of configurations on a weiqi board will never exceed this number. This is indeed the ?final answer.

【Notes】

[1] Yi Xing: Original name Zhang Sui (673 – 727 AD), astronomer, mathematician, inventor and Buddhist monk of the Tang Dynasty.

[2] The character “稊” ti should be written as “秭” zi.

[3] The correct number should be 150,094,635,296,999,121.

[4] Shen Kuo gave the value ; however, the more accurate value is .

[5] The correct number should be 1,350,851,717,672,992,089.

Article by Tao Steven Zheng（鄭濤）