# Time 1, Exercise 1: how the three natural cycles fit together
# Find year / month, how far 12 and 13 months miss a year, and which whole
# number of years lines up best with a whole number of months (up to 39 years).

YEAR = 365.2422        # tropical year, days
MONTH = 29.530589      # synodic month, days

print("year / month:", round(YEAR / MONTH, 3))
for n in (12, 13):
    total = n * MONTH
    print(n, "months =", round(total, 3), "days; year minus months =", round(YEAR - total, 3))

best = None
for years in range(1, 40):
    months = round(years * YEAR / MONTH)
    gap = abs(months * MONTH - years * YEAR)
    if best is None or gap < best[0] - 0.0001:
        best = (gap, years, months)
        print(years, "years ~", months, "months, gap", round(gap, 3), "days")

# Output when run:
# year / month: 12.368
# 12 months = 354.367 days; year minus months = 10.875
# 13 months = 383.898 days; year minus months = -18.655
# 1 years ~ 12 months, gap 10.875 days
# 2 years ~ 25 months, gap 7.78 days
# 3 years ~ 37 months, gap 3.095 days
# 8 years ~ 99 months, gap 1.591 days
# 11 years ~ 136 months, gap 1.504 days
# 19 years ~ 235 months, gap 0.087 days
#
# Reading the output: a year holds 12.368 months, so 12 months fall 10.875
# days short and 13 overshoot by 18.655 days. Neither works. Searching up to
# 39 years, each new record for the best fit is printed; the winner is 19
# years = 235 months, off by only 0.087 days (about two hours). This 19-year
# match was noticed by ancient astronomers; Time 3 returns to it.