How do you do an I Ching coin toss with 3 coins?

In the traditional three-coin method, assign heads a value of 3 and tails a value of 2. Shake and roll three coins together six times, recording each line from the bottom up: three tails equal 6 (changing yin), two tails and one head equal 7 (stable yang), two heads and one tail equal 8 (stable yin), and three heads equal 9 (changing yang).

How does the two-number I Ching casting method work?

In the classical Plum Blossom (Meihua) two-number method, the first integer modulo 8 selects the upper trigram (1=Qian to 8=Kun), the second integer modulo 8 selects the lower trigram, and the sum of both integers plus the solar branch hour (1–12) modulo 6 determines the single moving line. Because the arithmetic is deterministic, identical inputs in the same solar window always reproduce the exact same hexagram.

Can I use regular coins for an I Ching online cast or physical toss?

Yes. Any three identical coins work for a manual toss as long as you consistently treat heads as 3 and tails as 2, or you can use our online cast button above to simulate six three-coin rolls immediately.

Is this I Ching generator completely free without registration?

Yes. Every 6-line cast mode, changing-line transition, and all 64 classical hexagram guides run free in your browser with no sign-up required.

Free I Ching Generator: Online 64 Hexagram Calculator and Cast Tool

Q: How does an I Ching generator calculate changing lines?

Instead of picking a single random integer between 1 and 64, an authentic I Ching generator performs six independent line operations from bottom (Line 1) to top (Line 6). Each toss produces a line value of 6 (old yin, changing to yang), 7 (young yang, stable), 8 (young yin, stable), or 9 (old yang, changing to yin). Any line that scores 6 or 9 flips its binary bit to form the relating hexagram.

Q: How do the 8 trigrams combine inside a 6-line hexagram?

Each of the 8 trigrams is a 3-bit binary figure made of solid (1) or broken (0) lines. Stacking an upper outer trigram (Lines 4–6) above a lower inner trigram (Lines 1–3) creates a 6-bit figure with 64 unique hexagrams.

This interactive I Ching generator unites three classical casting inputs under one rule engine: an authentic three-coin toss simulator, a 50-stalk yarrow algorithm, and a reproducible two-number hexagram calculator. Unlike basic scripts that pick a flat integer between 1 and 64, our I Ching generator executes six bottom-to-top line operations (Line 1 to Line 6) so every online cast preserves changing lines (6 and 9). Once your six lines are set, jump directly to our I Ching hexagram meanings for all 64 hexagrams and 8×8 hexagram chart.

Interactive I Ching Coin Toss and 6-Line Hexagram Calculator

Choose your casting input below, then click Toss 3 Coins (Full 6-Line Cast) or enter two integers for a deterministic cast — 100% free with no account required. Whether you prefer a traditional I Ching coin toss (3:3:1:1 for 7:8:6:9, or 1:3:3:1 for 6:7:8:9), the 50-stalk yarrow division (1:3:5:7), or our reproducible two-number mode (Mod-8 trigrams + Mod-6 line), this I Ching generator and I Ching calculator computes your primary and relating figures automatically.

Casting Input: ●● 2 of 2 Free Casts Left

Two-Number Coordinate Cast (1–999): Num 1 (Upper): Num 2 (Lower): Audit: 38 mod 8 = 6 (Kan ☵) · 72 mod 8 = 8 (Kun ☷) · (38+72) mod 6 = Line 2

6-Line Stack (Bottom Line 1 to Top Line 6) Awaiting Cast (0/6)

L6 (Top) Awaiting cast...

L5 Awaiting cast...

L4 Awaiting cast...

L3 Awaiting cast...

L2 Awaiting cast...

L1 (Base) Awaiting cast...

Hexagram Calculation Tray

Ready to Cast — Select Mode & Generate Above

Step 1: Choose binomial coin, 50-stalk yarrow, two-number, or uniform 6-bit mode.

Step 2: Click Generate Full 6-Line Cast or compute by two integers above.

Moving lines (6 and 9) automatically derive your relating figure.

Free Online Cast: Zero sign-up required for 6-line generation and all 64 guides.

Browse 64 Hexagrams → Changing Lines Lookup →

I Ching generator and 3-coin toss calculator showing 6-bit binary hexagram matrix

▶ Watch 2K Lecture: 6-Bit XOR State Machine (2:47)

EP.02 · Click-to-play 2K math visualization YouTube ↗

How Our I Ching Generator Works: 3 Coins, Yarrow Stalks, and Two Numbers

Whether you toss physical I Ching coins, divide yarrow stalks, or input two numbers in the I Ching generator above, all three methods feed into the same six-line rule engine from Line 1 (base) to Line 6 (top).

1. The 3-Coin Value Rule (Heads = 3, Tails = 2)

In a classical iching coin toss, the yang side scores 3 and the yin side scores 2. Rolling three coins produces 2³ = 8 microstates across four line sums — a 3:3:1:1 ratio grouped by stability (7:8:6:9), or 1:3:3:1 ordered by line values (6:7:8:9):

Three Tails (2+2+2 = 6 · Old Yin): 1/8 (12.5%) — changing yin line flipping to yang (0 → 1).
Two Tails, One Head (2+2+3 = 7 · Young Yang): 3/8 (37.5%) — stable solid yang line (1).
Two Heads, One Tail (3+3+2 = 8 · Young Yin): 3/8 (37.5%) — stable broken yin line (0).
Three Heads (3+3+3 = 9 · Old Yang): 1/8 (12.5%) — changing yang line flipping to yin (1 → 0).

2. Why Our I Ching Generator Preserves All 4,096 Transitions

Basic pickers that draw one integer from 1 to 64 omit changing lines. Because any of the 64 primary figures in the Book of Changes can transform into any of the 64 figures, our I Ching generator models all 64 × 64 = 4,096 cast transitions.

3. Comparing the 3-Coin Roll vs. the 50-Stalk Yarrow Calculator

Recorded in the Great Treatise, the 50-stalk yarrow method sets one observer stalk aside and divides 49 stalks three times per line modulo 4. In our I Ching generator, yarrow mode yields an asymmetric 1:3:5:7 ratio (for lines 6:9:7:8) compared with the three-coin 1:3:3:1 ratio:

Line Sum	Classical Line Name	Binary Bit	3-Coin Toss Probability	Yarrow Stalk Probability	Line Behavior in Your Online Cast
6	Old Yin (Lao Yin)	`0 → 1`	2 / 16 (12.50%)	1 / 16 (6.25%)	Changing Yin: read this line statement; bit flips from `0` to `1`.
7	Young Yang (Shao Yang)	`1`	6 / 16 (37.50%)	5 / 16 (31.25%)	Stable Yang: structural solid line; remains `1`.
8	Young Yin (Shao Yin)	`0`	6 / 16 (37.50%)	7 / 16 (43.75%)	Stable Yin: structural broken line; remains `0`.
9	Old Yang (Lao Yang)	`1 → 0`	2 / 16 (12.50%)	3 / 16 (18.75%)	Changing Yang: read this line statement; bit flips from `1` to `0`.

4. The Two-Number Deterministic Method (Auditable Mod-8 & Mod-6 Engine)

For readers who prefer reproducible math over random draws, our I Ching generator and decision sandbox implement Shao Yong's Plum Blossom two-number rule using two integers (N₁, N₂) and the solar hour index (H ∈ 1..12):

Upper Trigram (Lines 4–6): N₁ mod 8 (0 → 8) in the Early Heaven order (1=Qian, 2=Dui, 3=Li, 4=Zhen, 5=Xun, 6=Kan, 7=Gen, 8=Kun).
Lower Trigram (Lines 1–3): N₂ mod 8 against the same index to stack your six-line figure.
Moving Line: (N₁ + N₂ + H) mod 6 (0 → 6) sets the single active moving line (9 or 6) for hand-auditable casts.

How to Read Your I Ching Online Cast in 3 Steps

To turn the output of this I Ching generator into a structured reading, follow three steps:

Read the Primary Hexagram: Lines 1–3 form the lower trigram and Lines 4–6 form the upper trigram; verify your pair in the 8×8 I Ching hexagram chart.
Isolate Changing Lines (6 and 9): Stable lines (7 and 8) stay silent; when lines score 6 or 9 in your I Ching coin toss or two-number cast, apply Zhu Xi's changing lines rules.
Compare the Relating Hexagram: Flipping every 6 and 9 yields your relating figure. Inspect it in our I Ching hexagram meanings lookup, decision sandbox, or Yes or No Matrix.

Leibniz Binary Logic and the Eight Trigrams Matrix

As Gottfried Leibniz showed in 1703, mapping yin to 0 and yang to 1 turns the eight trigrams into 3-bit vectors (000–111) and all 64 figures in our I Ching generator into 6-bit words (000000–111111).

Symbol	Trigram	3-Bit	Image	Attribute	Pure Hexagram
☰	Qian	`111`	Heaven	Creative drive	#1 Qian
☷	Kun	`000`	Earth	Receptive execution	#2 Kun
☳	Zhen	`100`	Thunder	Kinetic impulse	#51 Zhen
☴	Xun	`011`	Wind	Adaptation	#57 Xun
☵	Kan	`010`	Water	Core integrity	#29 Kan
☲	Li	`101`	Fire	Clarity	#30 Li
☶	Gen	`001`	Mountain	Stillness	#52 Gen
☱	Dui	`110`	Lake	Negotiation	#58 Dui

All 64 Hexagrams Quick Index and Generator FAQs

Click any figure below to open its full Judgment, Image, and six line readings, or view all I Ching hexagram meanings and the 64 hexagrams chart on a single page:

#1 Qian (The Creative) #2 Kun (The Receptive) #3 Zhun (Difficulty at Beginning) #4 Meng (Youthful Folly) #5 Xu (Waiting) #6 Song (Conflict) #7 Shi (The Army) #8 Bi (Union) #9 Xiao Xu (Small Restraint) #10 Lu (Treading) #11 Tai (Peace) #12 Pi (Standstill) #13 Tong Ren (Fellowship) #14 Da You (Great Possession) #15 Qian (Humility) #16 Yu (Enthusiasm) #17 Sui (Following) #18 Gu (Decay) #19 Lin (Approach) #20 Guan (Contemplation) #21 Shi He (Biting Through) #22 Bi (Grace) #23 Bo (Splitting Apart) #24 Fu (Return) #25 Wu Wang (Innocence) #26 Da Xu (Great Accumulation) #27 Yi (Nourishment) #28 Da Guo (Preponderance) #29 Kan (Abyssal Water) #30 Li (Clinging Fire) #31 Xian (Mutual Attraction) #32 Heng (Duration) #33 Dun (Retreat) #34 Da Zhuang (Great Power) #35 Jin (Progress) #36 Ming Yi (Darkening Light) #37 Jia Ren (The Family) #38 Kui (Opposition) #39 Jian (Obstruction) #40 Xie (Deliverance) #41 Sun (Decrease) #42 Yi (Increase) #43 Guai (Breakthrough) #44 Gou (Coming to Meet) #45 Cui (Gathering) #46 Sheng (Pushing Upward) #47 Kun (Oppression) #48 Jing (The Well) #49 Ge (Revolution) #50 Ding (The Cauldron) #51 Zhen (Arousing Thunder) #52 Gen (Keeping Still) #53 Jian (Gradual Progress) #54 Gui Mei (Marrying Maiden) #55 Feng (Abundance) #56 Lu (The Wanderer) #57 Xun (Gentle Wind) #58 Dui (Joyous Lake) #59 Huan (Dispersion) #60 Jie (Limitation) #61 Zhong Fu (Inner Truth) #62 Xiao Guo (Small Preponderance) #63 Ji Ji (After Completion) #64 Wei Ji (Before Completion)

How does an I Ching generator calculate changing lines?

Instead of drawing one flat integer from 1 to 64, a true I Ching generator computes six line states (6, 7, 8, or 9) from bottom to top, flipping any 6 or 9 into the relating figure.

How do you cast with three coins at home?

Assign heads = 3 and tails = 2 across six throws in our dedicated I Ching coins simulator: 6 and 9 move, while 7 and 8 stay stable.

How does the two-number casting method work?

Enter two positive integers and a solar hour (1–12): N₁ mod 8 sets the upper trigram, N₂ mod 8 sets the lower trigram, and (N₁+N₂+H) mod 6 sets the moving line in our I Ching generator.

What is the difference between yarrow-stalk and uniform binary mode?

Yarrow mode models the classical 1:3:5:7 moving-line odds, whereas uniform 6-bit binary mode samples static 64-state vectors without moving lines.

How do the 8 trigrams combine inside a 6-line hexagram?

Stacking a 3-bit upper outer trigram (Lines 4–6) above a 3-bit lower inner trigram (Lines 1–3) forms 8 × 8 = 64 unique hexagrams.

Is this I Ching generator free without registration?

Yes — all casting modes, changing-line rules, and 64 classical guides run free in your browser with no account required.