The ritual of the 49 yarrow stalks
Where 1, 3, 5 and 7 sixteenths come from
Cast the 49 stalks line by line, or simulate thousands and watch the fractions converge.
The experiment on coins against yarrow takes as given that the yarrow splits its lines with probabilities 3/16, 7/16, 5/16 and 1/16. Here we do not assume them: we derive them from the ancient procedure. Of 50 stalks one is set aside (it stays out of the whole rite) and the remaining 49 go through 3 operations: divide the bundle in two, take one stalk from the right pile, and count each pile 4 by 4, setting the remainders aside. What is left at the end, divided by 4, gives the value of the line: 6, 7, 8 or 9.
It starts at 50; one is set aside before the rite and never returns: 49 remain.
The exact derivation rests on an assumption: at each division, the remainder of the left pile when counting 4 by 4 (1, 2, 3 or 4) is taken as equiprobable. It is the classic idealised model of the rite. With real hands, which split the bundle near the middle, the distribution of remainders is approximately uniform but not exact: other splitting assumptions give other numbers. Everything that follows holds under that assumption, which the interactive walkthrough above also uses.
The first operation sets aside 5 stalks with probability 3/4, or 9 with 1/4. The other two set aside 4 or 8 with probability 1/2 each. The resulting branches, grouped by their final value, give exactly 3/16, 7/16, 5/16 and 1/16.
Comparison by state: derived here, used by the probabilities experiment, and simulated.
| State | Derived | Oracle table | Simulated |
|---|---|---|---|
| 9 · old yang ● | 3/16 | 3/16 = | … (theory 18.75%) |
| 8 · young yin | 7/16 | 7/16 = | … (theory 43.75%) |
| 7 · young yang | 5/16 | 5/16 = | … (theory 31.25%) |
| 6 · old yin ● | 1/16 | 1/16 = | … (theory 6.25%) |
The table that the experiment coins against yarrow takes as given matches, state by state, the derivation of this rite.
The exact enumeration of the 4^3 = 64 cases of the ritual of the 49 yarrow stalks gives 3/16, 5/16, 7/16 and 1/16, identical to the table of the probabilities experiment.
verificar_milenramaThe types of claim and the full bibliography (APA) are in Foundations.