Discussing an Interesting Phenomenon in Plum Blossom Divination Caused by Deduction Errors

The previous article<<Solved the Problem of Using True Random Numbers for Casting Hexagrams>>thought that the problem of uniform distribution modulo 9 on computers had been solved. However, through practice, it was found that this method not only fails to produce a uniform distribution but instead results in a normal distribution, because two random numbers were added together. Therefore, how to ensure a strictly uniform probability distribution for the Nine Palaces remains unsolved for now.

The cumulative sum of continuous, distinct single-probability events generally follows a normal distribution, satisfying the Central Limit Theorem.

To put it more simply, picking a random number from 0 to 8 yields a uniform distribution. However, if you first pick one number from 0 to 4, and then pick another from 0 to 4, the cumulative sum of these two numbers follows a normal distribution, which is different from the probability distribution of picking any single number directly from 0 to 8.

This is because there are 9 numbers in total from 0 to 8. Although the sum of two picks from 0 to 4 appears to range from 0 to 8, it actually involves selecting from 10 possible outcomes (0+0 to 4+4), causing probability overlap, which leads to the appearance of a normal distribution.

 For example, using the above method to first pick a number from 0 to 4, then pick another, and sum them to determine the hexagram, performing this 1,000 times results in the following distribution:

 Qian 63
 Dui 112
 Li 178
 Zhen 235
 Xun 192
 Kan 124
 Gen 60
 Kun 36

So, does such a phenomenon exist in hexagram casting, such as in Plum Blossom Divination? Let's analyze it specifically.

Taking the method of casting hexagrams by reporting numbers as an example: report two numbers, such as 1 and 2, to obtain the Qian and Dui hexagrams. Then, 1+2=3, indicating that the third line moves, thus generating a changed hexagram.

The Plum Blossom hexagrams generated by reporting numbers follow a normal distribution. Simulating 10,000 random number reports, since two hexagrams are produced each time, random statistics reveal the following situation for the primary hexagram:

Upper Hexagram: Qian 1283, Dui 1259, Li 1216, Zhen 1261, Xun 1266, Kan 1296, Gen 1216, Kun 1203
Lower Hexagram: Qian:1245, Dui:1244, Li:1268, Zhen:1327, Xun:1222, Kan:1260, Gen:1179, Kun:1255

Combinations: Qian-Qian:158, Qian-Dui:177, Qian-Li:155, Qian-Zhen:149, Qian-Xun:152, Qian-Kan:166, Qian-Gen:158, Qian-Kun:168
Dui-Qian:154, Dui-Dui:145, Dui-Li:172, Dui-Zhen:180, Dui-Xun:153, Dui-Kan:142, Dui-Gen:158, Dui-Kun:155
Li-Qian:147, Li-Dui:155, Li-Li:155, Li-Zhen:166, Li-Xun:157, Li-Kan:155, Li-Gen:136, Li-Kun:145
Zhen-Qian:157, Zhen-Dui:153, Zhen-Li:162, Zhen-Zhen:189, Zhen-Xun:161, Zhen-Kan:164, Zhen-Gen:132, Zhen-Kun:143
Xun-Qian:151, Xun-Dui:164, Xun-Li:158, Xun-Zhen:156, Xun-Xun:149, Xun-Kan:172, Xun-Gen:142, Xun-Kun:174
Kan-Qian:178, Kan-Dui:152, Kan-Li:161, Kan-Zhen:162, Kan-Xun:148, Kan-Kan:164, Kan-Gen:166, Kan-Kun:165
Gen-Qian:161, Gen-Dui:138, Gen-Li:157, Gen-Zhen:155, Gen-Xun:160, Gen-Kan:145, Gen-Gen:150, Gen-Kun:150
Kun-Qian:139, Kun-Dui:160, Kun-Li:148, Kun-Zhen:170, Kun-Xun:142, Kun-Kan:152, Kun-Gen:137, Kun-Kun:155

Obviously, the probability of the primary hexagram appearing is uniformly distributed, ensuring that each of the 64 hexagrams has the possibility of appearing. However, it is important to note that with the changed hexagram, the situation is different because the changed hexagram involves the accumulation of numerical values.

Here we simulate again. To ensure a uniform distribution of all 64 hexagrams in the primary hexagram, while allowing the moving lines from one to six to be complete (totaling 386 lines), we directly use 384 as the sample space for statistics.

 A change in the first line of a hexagram is equivalent to flipping the least significant bit; a change in the second line is equivalent to flipping the second bit, and so on.

Primary Hexagram:

Upper Hexagram: Qian:1274, Dui:1252, Li:1200, Zhen:1217, Xun:1261, Kan:1215, Gen:1304, Kun:1277
Lower Hexagram: Qian:1280, Dui:1232, Li:1204, Zhen:1258, Xun:1287, Kan:1232, Gen:1250, Kun:1257

Qian-Qian:161, Qian-Dui:169, Qian-Li:169, Qian-Zhen:164, Qian-Xun:138, Qian-Kan:144, Qian-Gen:162, Qian-Kun:167
Dui-Qian:174, Dui-Dui:146, Dui-Li:167, Dui-Zhen:159, Dui-Xun:155, Dui-Kan:169, Dui-Gen:125, Dui-Kun:157
Li-Qian:145, Li-Dui:161, Li-Li:147, Li-Zhen:141, Li-Xun:148, Li-Kan:151, Li-Gen:164, Li-Kun:143
Zhen-Qian:161, Zhen-Dui:143, Zhen-Li:138, Zhen-Zhen:157, Zhen-Xun:145, Zhen-Kan:164, Zhen-Gen:152, Zhen-Kun:157
Xun-Qian:151, Xun-Dui:153, Xun-Li:145, Xun-Zhen:160, Xun-Xun:180, Xun-Kan:167, Xun-Gen:157, Xun-Kun:148
Kan-Qian:148, Kan-Dui:134, Kan-Li:161, Kan-Zhen:156, Kan-Xun:168, Kan-Kan:130, Kan-Gen:153, Kan-Kun:165
Gen-Qian:175, Gen-Dui:172, Gen-Li:128, Gen-Zhen:166, Gen-Xun:184, Gen-Kan:147, Gen-Gen:174, Gen-Kun:158
Kun-Qian:165, Kun-Dui:154, Kun-Li:149, Kun-Zhen:155, Kun-Xun:169, Kun-Kan:160, Kun-Gen:163, Kun-Kun:162

Changed Hexagram:

Upper Hexagram: Qian:1248, Dui:1213, Li:1243, Zhen:1238, Xun:1308, Kan:1226, Gen:1248, Kun:1276
Lower Hexagram: Qian:1295, Dui:1254, Li:1189, Zhen:1223, Xun:1271, Kan:1260, Gen:1246, Kun:1262

Qian-Qian:174, Qian-Dui:174, Qian-Li:164, Qian-Zhen:171, Qian-Xun:168, Qian-Kan:158, Qian-Gen:134, Qian-Kun:152
Dui-Qian:157, Dui-Dui:134, Dui-Li:170, Dui-Zhen:171, Dui-Xun:158, Dui-Kan:158, Dui-Gen:141, Dui-Kun:165
Li-Qian:131, Li-Dui:126, Li-Li:148, Li-Zhen:166, Li-Xun:159, Li-Kan:155, Li-Gen:152, Li-Kun:152
Zhen-Qian:153, Zhen-Dui:145, Zhen-Li:167, Zhen-Zhen:149, Zhen-Xun:164, Zhen-Kan:158, Zhen-Gen:141, Zhen-Kun:146
Xun-Qian:159, Xun-Dui:153, Xun-Li:157, Xun-Zhen:140, Xun-Xun:159, Xun-Kan:165, Xun-Gen:175, Xun-Kun:163
Kan-Qian:162, Kan-Dui:168, Kan-Li:146, Kan-Zhen:153, Kan-Xun:150, Kan-Kan:149, Kan-Gen:154, Kan-Kun:178
Gen-Qian:158, Gen-Dui:155, Gen-Li:124, Gen-Zhen:138, Gen-Xun:178, Gen-Kan:148, Gen-Gen:193, Gen-Kun:152
Kun-Qian:154, Kun-Dui:158, Kun-Li:167, Kun-Zhen:150, Kun-Xun:172, Kun-Kan:135, Kun-Gen:158, Kun-Kun:168

It can be seen that although the upper and lower hexagrams are added together, the situation of a normal distribution caused by probability overlap does not occur. In other words, the method of casting hexagrams in Plum Blossom Divination avoids leading to a normal distribution, so the method of casting hexagrams in Plum Blossom Divination is extremely scientific.