Guide
Three coins do not produce four lines equally.
A three-coin cast yields line values 6, 7, 8, and 9 with probabilities 1/8, 3/8, 3/8, 1/8 — not the uniform 1/4 each that some digital tools assume. Changing lines (6 and 9) are equally likely at 1/8 each: a 1/4 chance per line, about 1.5 moving lines in a six-line cast.
The derivation
Each coin lands on one of two faces, valued 2 or 3, each equally likely. Three coins give 2³ = 8 equally likely combinations, and the line value is their sum (minimum 2+2+2 = 6, maximum 3+3+3 = 9):
| Line | Type | Changing? | Ways (of 8) | Probability |
|---|---|---|---|---|
| 6 | old yin (老阴) | yes → yang | 1 | 1/8 = 12.5% |
| 7 | young yang (少阳) | no | 3 | 3/8 = 37.5% |
| 8 | young yin (少阴) | no | 3 | 3/8 = 37.5% |
| 9 | old yang (老阳) | yes → yin | 1 | 1/8 = 12.5% |
The counts 1 : 3 : 3 : 1 are the binomial coefficients of (a + b)³. That is the whole rule — everything below is what it implies.
Coins vs yarrow stalks vs the naive 1/4
The older yarrow-stalk method produces a different distribution, and a flat “pick 6/7/8/9 each at 1/4” matches neither:
| Outcome | Three coins | Yarrow stalks | Naive uniform |
|---|---|---|---|
| 6 — old yin (changing) | 1/8 | 1/16 | 1/4 |
| 7 — young yang | 3/8 | 5/16 | 1/4 |
| 8 — young yin | 3/8 | 7/16 | 1/4 |
| 9 — old yang (changing) | 1/8 | 3/16 | 1/4 |
| Any changing line (6 or 9) | 1/4 | 1/4 | 1/2 |
The yarrow fractions (1 : 5 : 7 : 3) are the theoretical values for the standard 50-stalk ritual with unbiased heap division. The naive uniform method doubles the changing-line rate, from 1/4 to 1/2 — you would get moving lines twice as often as the standard coin or yarrow cast allows.
What most explanations get wrong
It is widely repeated that yarrow stalks produce “fewer moving lines” than coins. That is not true. Both methods give a 1/4 total changing-line rate (coins 1/8 + 1/8; yarrow 1/16 + 3/16) and an expected 1.5 moving lines per cast. What actually differs is which kind of line moves:
- Coins are symmetric. A changing yang (9) and a changing yin (6) are equally likely, 1/8 each.
- Yarrow is asymmetric. A changing yang (9, 3/16) is three times as likely as a changing yin (6, 1/16); stable young yin (8, 7/16) is the single most common outcome.
So the genuine coins-versus-yarrow difference is the yin/yang balance of change, not the amount of it.
Why YiGram uses the real distribution
YiGram casts with the true three-coin distribution (1/8, 3/8, 3/8, 1/8), so the rate at which you draw moving lines — and therefore a transformed hexagram — matches a physical cast rather than a shortcut. The casting rule is part of the open, versioned ruleset: read it on GitHub, or see how the three layers fit together in the methodology.
Questions
- Are the I Ching line values 6, 7, 8, and 9 equally likely?
- No. With three coins they occur with probability 1/8, 3/8, 3/8, 1/8 — the binomial 1:3:3:1 ratio, not a flat 1/4 each.
- How often do you get a changing line?
- A line changes when it is a 6 or a 9, which is 1/8 + 1/8 = 1/4 per line — about 1.5 moving lines in a six-line cast. The total is the same for coins and yarrow stalks.
- Do yarrow stalks give the same odds as three coins?
- The total changing-line rate is identical (1/4), but yarrow is asymmetric: a changing yang (9) is three times as likely as a changing yin (6). Three coins make the two equally likely.
- Is the three-coin method less accurate than yarrow?
- Neither is 'more accurate' — they encode different probabilities. What is wrong is the naive 'pick 6/7/8/9 each at 1/4' some digital tools use: it matches neither method and doubles the changing-line rate to 1/2.