I Ching Divination Mechanics

A yang line is a 1 and a yin line is a 0, so six of them are a number from 0 to 63 — which is what the 6-Bit Binary readout shows, and why there are exactly 64 hexagrams and not some other count. This is not a modern re-reading. Leibniz published his paper on binary arithmetic in 1703 and part of its title runs sur ce qu'elle donne le sens des anciennes figures de Fohy: on how binary gives the sense of the ancient figures of Fu Xi. The Jesuit Joachim Bouvet had sent him the hexagram arrangement from Beijing and he recognised his own notation in it. Eight trigrams are the eight three-bit numbers; sixty-four hexagrams are the six-bit ones.

Coins give the four line values 1/8, 3/8, 3/8, 1/8. Yarrow stalks give 1/16, 5/16, 7/16, 3/16. They are described almost everywhere as interchangeable, and they are not: under the stalks a moving line is yang three quarters of the time, where coins make it yang half the time. The stalk ritual leans, hard, toward yang collapsing into yin. Switching between the two methods above moves the P(Yang | Moving) readout from 50.0% to 75.0% and shifts every green marker on the chart. Whoever reached for coins in the twentieth century because they were quicker also removed a bias the ritual had carried for three thousand years, and as far as the record shows nobody mentioned it.

Both methods make a line settled exactly 3/4 of the time. Both expect exactly 1.5 moving lines per hexagram. Both give a completely unchanging hexagram (3/4)&sup6; = 729/4096 = 17.798% of the time. Count how many lines move and the two are indistinguishable however long you cast; the whole of their difference is which way those lines go. Hexagram 1 turning into hexagram 2, all six yang lines moving at once, is 1 in 262,144 by coins and 1 in 23,014 by stalks — likelier by a factor of 729/64 — while the same journey in reverse is 64 times rarer. Two rituals that agree on how much the world changes and disagree on which direction it changes in.

Those sixteenths follow from assuming that each heap, after the fours are counted out, is equally likely to leave 1, 2, 3 or 4 stalks. That is an assumption about a hand dividing a bundle of 49, and it is not the only natural one. Assume instead that the hand is equally likely to cut anywhere along the bundle, enumerate every split, and the figures come out 5.174 / 28.874 / 44.836 / 21.117% — which is the third button above. Look at what that breaks: the 3/4 settled share the two classical methods share exactly becomes 73.71%, and 1.5 expected moving lines becomes 1.5774. Neither model is the truth about a real person with real stalks. The point is that the textbook numbers are the consequence of a modelling choice that is almost never written down beside them.