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    OPEN STUDY / 03

    Stems and branches

    Explore the sixty-pair cycle with forward and backward steps, wraparound, worked calculations and four self-checks.

    About 12 min
    Study notes & citation
    Saved automatically in this browser0 / 12000

    Stems and branches. Institute of Mantic Arts. https://www.manticarts.com/en/learning/stems-branches

    STUDY / OBJECTIVE

    Distinguish sequence, counting, parity and the time unit being labeled.

    01 / STUDY

    A sequence and a date are separate questions

    The cycle first provides repeating labels. Knowing a label's position does not identify a calendar date. That requires a reliable reference date, time unit and counting method. The same pair recurs in different rounds, so preserve a larger temporal context.

    02 / STUDY

    Position numbers and steps moved

    A table usually numbers positions one to sixty, while calculations often start at zero. With Jia-Zi at index zero, one step reaches Yi-Chou at index one. Inclusive counting calls these items one and two. Both conventions work, provided the record never switches between them silently.

    03 / STUDY

    Pairs absent from the cycle

    Synchronous movement keeps the stem and branch positions at matching parity. Although arbitrary pairing gives 120 possibilities, this procedure visits only sixty. When a pair is absent, examine transcription, indexing and the material's purpose before silently replacing it with a nearby valid pair.

    WORKED EXAMPLE

    Check the sequence before attaching a unit

    The exercise starts at Jia-Zi and moves twelve steps; no actual date is specified.

    1. Twelve stem steps equal one full round plus two, reaching Bing.

    2. Twelve branch steps complete a round and return to Zi. The resulting pair is Bing-Zi.

    3. This is position thirteen in the table. Record twelve sequence steps, without labeling them years or days.

    A step count becomes a time interval only when the input unit is established.

    PRACTICE / REFLECTION

    Exercises and answers

    01Why do neither ten nor twelve steps reset both sequences?

    Read the explanation

    Ten resets the stems only; twelve resets the branches only. A joint return requires a multiple of both, with sixty the smallest positive value.

    02Can “Jia-Zi day” identify a Gregorian date by itself?

    Read the explanation

    No. The label repeats. Supply a date range and reliable calendar correspondence, retaining alternatives when the evidence is incomplete.

    A distinction to retain

    Year, month, day and hour labels cannot be converted into one another with this table alone; each application has additional origins and boundaries.

    Unit output

    Three forward/backward traces, an explanation of an invalid pair and an input card stating its time unit.

    REFERENCES

    Reading references

    References reviewed: 2026-10-24. Examples, record templates and explanations are editorial teaching material. Fictional inputs are not historical events or efficacy evidence. Citations imply no affiliation.

    SEQUENCE / PRACTICE

    The sixty-pair cycle, step by step

    Place the ten stems and twelve branches in their fixed orders. Begin at Jia and Zi, moving both sequences by one place at each step and wrapping each independently. The resulting sixty pairs can be checked against Table 2 in the Hong Kong Observatory reference.

    The first joint return takes the least common multiple of 10 and 12: sixty steps. Odd positions pair with odd positions, and even positions pair with even positions. An arbitrary stem and branch therefore need not form a pair in this sequence.

    Positive steps move forward; negative steps move backward. Zero keeps the origin. Range: −9999 to 9999. The origin is not counted as a step.

    Result 01 / 60甲子Jiǎ–Zǐ

    癸亥 + 1 → 甲子

    Ten stems

    1. 1甲
    2. 2乙
    3. 3丙
    4. 4丁
    5. 5戊
    6. 6己
    7. 7庚
    8. 8辛
    9. 9壬
    10. 10癸

    Twelve branches

    1. 1子
    2. 2丑
    3. 3寅
    4. 4卯
    5. 5辰
    6. 6巳
    7. 7午
    8. 8未
    9. 9申
    10. 10酉
    11. 11戌
    12. 12亥
    Calculation trace

    Table positions run from 1 to 60; calculation indices run from 0 to 59. Keep the remainder in 0–59, then add 1 to recover the table position.

    Starting position
    60 − 1 = 59
    Steps to move
    59 + (1) = 60
    mod 60
    60 = 60 × (1) + 0
    Table position
    0 + 1 = 1 · 甲子

    The sixty-pair cycle

    A gold outline marks the origin; dark green marks the result.

    1. 01甲子Jiǎ–Zǐ
    2. 02乙丑Yǐ–Chǒu
    3. 03丙寅Bǐng–Yín
    4. 04丁卯Dīng–Mǎo
    5. 05戊辰Wù–Chén
    6. 06己巳Jǐ–Sì
    7. 07庚午Gēng–Wǔ
    8. 08辛未Xīn–Wèi
    9. 09壬申Rén–Shēn
    10. 10癸酉Guǐ–Yǒu
    11. 11甲戌Jiǎ–Xū
    12. 12乙亥Yǐ–Hài
    13. 13丙子Bǐng–Zǐ
    14. 14丁丑Dīng–Chǒu
    15. 15戊寅Wù–Yín
    16. 16己卯Jǐ–Mǎo
    17. 17庚辰Gēng–Chén
    18. 18辛巳Xīn–Sì
    19. 19壬午Rén–Wǔ
    20. 20癸未Guǐ–Wèi
    21. 21甲申Jiǎ–Shēn
    22. 22乙酉Yǐ–Yǒu
    23. 23丙戌Bǐng–Xū
    24. 24丁亥Dīng–Hài
    25. 25戊子Wù–Zǐ
    26. 26己丑Jǐ–Chǒu
    27. 27庚寅Gēng–Yín
    28. 28辛卯Xīn–Mǎo
    29. 29壬辰Rén–Chén
    30. 30癸巳Guǐ–Sì
    31. 31甲午Jiǎ–Wǔ
    32. 32乙未Yǐ–Wèi
    33. 33丙申Bǐng–Shēn
    34. 34丁酉Dīng–Yǒu
    35. 35戊戌Wù–Xū
    36. 36己亥Jǐ–Hài
    37. 37庚子Gēng–Zǐ
    38. 38辛丑Xīn–Chǒu
    39. 39壬寅Rén–Yín
    40. 40癸卯Guǐ–Mǎo
    41. 41甲辰Jiǎ–Chén
    42. 42乙巳Yǐ–Sì
    43. 43丙午Bǐng–Wǔ
    44. 44丁未Dīng–Wèi
    45. 45戊申Wù–Shēn
    46. 46己酉Jǐ–Yǒu
    47. 47庚戌Gēng–Xū
    48. 48辛亥Xīn–Hài
    49. 49壬子Rén–Zǐ
    50. 50癸丑Guǐ–Chǒu
    51. 51甲寅Jiǎ–Yín
    52. 52乙卯Yǐ–Mǎo
    53. 53丙辰Bǐng–Chén
    54. 54丁巳Dīng–Sì
    55. 55戊午Wù–Wǔ
    56. 56己未Jǐ–Wèi
    57. 57庚申Gēng–Shēn
    58. 58辛酉Xīn–Yǒu
    59. 59壬戌Rén–Xū
    60. 60癸亥Guǐ–Hài

    Four checks of understanding

    01 · Move one step forward from position 60, Gui-Hai. Where do you arrive?

    Reference explanation

    Position 1, Jia-Zi. Gui wraps to Jia and Hai wraps to Zi. This identifies a sequence position; no year, month, day or hour has been assigned.

    02 · Do one backward step and 59 forward steps from Jia-Zi reach the same pair?

    Reference explanation

    Both reach position 60, Gui-Hai. Their paths differ by one full cycle. Equal endpoints do not mean equal distances, so retain the starting point and step count.

    03 · Why does Jia-Zi advanced by ten steps keep Jia but lose Zi?

    Reference explanation

    The ten-stem sequence completes a round, while the twelve-branch sequence reaches Xu. The result is position 11, Jia-Xu. Both sequences return together only after a common multiple of 10 and 12 steps.

    04 · How does the tenth item counted from Jia-Zi differ from moving ten steps from Jia-Zi?

    Reference explanation

    Counting Jia-Zi as item one requires nine moves to reach item ten, Gui-You. Moving ten steps reaches position eleven, Jia-Xu. Establish the counting convention before applying a rule.

    Record the starting pair, position, step count, counting convention and result. This lab handles sequence relations; converting an actual date also requires a calendar, a reference date and boundary conventions.

    Check the reference: Hong Kong Observatory, Table 2 ↗

    Sequence checked 2026-10-24. Calculations, interaction and exercises are editorial teaching material; they do not calculate civil dates or natal charts.

    Next: calendars and solar terms →

    Personal study record; not a grade, assessment or credential.

    Connected study paths

    Calendars: cycles, terms and boundariesCosmic boards: objects and procedures

    Editorial study material · 2026-10-24