I Ching Memory Wheels

Sixty-four binary digits can be arranged to represent all the hexagrams of the I Ching in an overlapping sequence.  The sequence wraps around, like beads on a circular string; one could use such a string of beads to select hexagrams.  I have illustrated the concept for one sequence below following the example originally found in the Abrahadabra.com forum; please see The Changes and the Bruijn sequence on the Clarity site and De Bruijn sequence on Wikipedia for more information.

0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001 0000001010001001011000011001101001110001111010101110110111111001

You can roll your own sequences with the following script.  n is the length of the digit strings which are contained in the sequence; for hexagrams, it is 6.

Think of the sequences as cyclic, like beads on a circular string; each sequence can start at any position.

The number of distinct cyclic but not reversed sequences is 2^2(n-1)-n.  Divide this number in half to get the number of distinct reversible circular strings of beads.

string length n = 3 4 6 | start with: 0 random

sequence length = 2ⁿ =   | distinct sequences = 2^2(n-1)-n =  

generate | clear

Below the list of binary sequences are the decimal sequence for the last one generated, and a graphical representation using broken lines for 0 and solid lines for 1 (like hexagrams, turned sideways for convenience).  Run your mouse over the graphic to see each contained string, with its decimal representation and some further information.  Note that the strings wrap around to the beginning.

• For string length 6 (hexagrams), the decimal representation for each trigram is included.

• For string length 3 (trigrams), two ways of generating hexagram lines are included.  The first assigns the value 3 to solid lines, and 2 to broken ones, like heads and tails in the traditional coin method.  In the second method (my favorite), one selects a line by selecting a bead; and if the following two beads are the same as the first, the line is moving.  It’s the simplest, and the probabilities are the same as for the traditional coin method.

• For string length 4, one selects a line and a string of 4 by selecting a bead; and if exactly 3 beads in the string of 4 are solid, the line is moving.  The probabilities here are those of the yarrow stalk method.

The latter two methods are explained and illustrated in more detail on the coins page.

I used the short sequences to illustrate the coin-like methods, but of course you can use a longer one; with the 64-bead sequence, you can do everything.  But if a tiny “stealth” string of 8 beads is more your thing, you can select hexagrams by selecting two trigrams, or lines using the traditional coin method.

This script itself can be used to generate hexagrams.  For example, set the string length to 3 and the start to random, generate 6 sequences, and look at the first 3 digits of each.  (My own preference is to use the first sequence as the bottom line.)

What about string length 12 for hexagram pairs, you ask?  Good question; but unfortunately such a sequence has little practial use as it is 4096 beads long (85 ⅓ feet of ¼" beads, the height of a 10-story building).  But you wouldn’t have to worry about running out of distinct sequences; there are 7,889,894,060,378,663,891,776,092,941,569,812,490,342,798,503,348,506,843,781,822,614,141,761,508,512,668,675,096,973,000,860,086,294,845,878,408,428,197,758,665,556,532,995,725,322,532,980,505,953,722,725,371,247,772,794,751,219,748,962,450,622,653,723,658,885,756,259,371,110,246,161,326,225,254,545,838,906,698,753,500,604,533,806,294,347,131,844,801,035,861,224,582,895,768,019,679,578,222,442,365,816,777,811,116,550,211,997,661,300,042,672,501,115,599,330,121,102,374,590,346,210,401,772,297,613,816,579,213,943,884,340,476,773,480,741,388,169,825,126,577,601,893,944,909,949,292,394,019,629,808,987,605,783,015,436,309,981,618,852,590,264,930,933,038,801,592,643,964,487,934,730,639,144,298,339,176,843,993,808,158,087,198,825,546,224,497,949,740,096,003,321,446,610,956,690,662,145,555,067,347,073,012,596,736 of them, many times the number of atoms in the universe.