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The sexagenary, or Gan-Zhi, year cycle pairs ten heavenly stems with twelve earthly branches, moving one place forward in each list for every year-name. The pair returns to Jia-Zi after 60 steps because 60 is the least common multiple of 10 and 12. This is an algorithm for year labels, not a complete method for calculating Chinese calendar dates.
How do ten stems and twelve branches produce 60 year names?
Number the ten stems from 0 to 9 and the twelve branches from 0 to 11. At step n, pair stem n modulo 10 with branch n modulo 12. Starting at step 0 gives Jia-Zi. Each next step advances both lists by one.
The pair repeats only when both indices return to their starting positions. That requires a number divisible by 10 and 12; the smallest positive number is 60. Consequently, the sequence contains 60 ordered pairs, not 10 × 12 = 120, because the two lists advance together rather than being combined in every possible pairing. The Hong Kong Observatory presents the ordered stem-branch sequence and its cycle table in “Heavenly Stems and Earthly Branches”; NASA also describes the sexagenary cycle in its calendar reference.
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The stems and branches each follow their own fixed order. The pair at each position uses the same position in both repeating lists:
stems = [Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui]
branches = [Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai]
for n in 0..59:
print(stems[n mod 10], branches[n mod 12])The first pair is Jia-Zi. The 60th listed pair is Gui-Hai; advancing once more wraps both lists to Jia-Zi. This procedure generates the cyclical labels, but it does not by itself identify which absolute year a label refers to.
Why doesn’t a cycle name identify a unique Gregorian year?
A repeating sequence gives a position within a cycle, not a unique year across all history. To map a name such as Jia-Zi to an absolute year, you need an external year count or a chosen epoch. NASA’s calendar reference notes that the cycle has no specific initial epoch for counting years, so the name alone cannot establish a unique Gregorian year.
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How does the cycle differ from the Chinese calendar?
The Gan-Zhi sequence labels years; it is not the algorithm that determines Chinese lunisolar month dates, month lengths, leap months, or Lunar New Year. The Hong Kong Observatory explains that Lunar New Year falls on the first day of the second new moon after the winter solstice, excluding a leap month from that count, and describes the calendar’s use of seven leap months in 19 years. See its explanation, “What year is it today?”.
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When does the Gan-Zhi year change?
The boundary depends on the convention being used. The Hong Kong Observatory changes the Gan-Zhi year on Lunar New Year, the first day of the first Chinese calendar month. It distinguishes this from “Spring Commences,” a solar term that can arrive earlier. These are different boundary events, not interchangeable dates; the Observatory’s stated practice should not be assumed to represent every astrological or calendrical tradition. Its discussion of the year boundary is in “What year is it today?”.
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| Boundary event | Convention described | What determines the date |
|---|---|---|
| Lunar New Year | Hong Kong Observatory’s Gan-Zhi year change | First day of the first Chinese calendar month; it changes date from year to year. |
| “Spring Commences” | Used as a boundary in some discussions | A solar term that can occur before Lunar New Year; the exact date varies by year. |
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