# Time 1, Exercise 3: a moon tally on a bone
# A tally maker notches one mark a day. Month length is 29.530589 days.
# 1) If every month is counted as 29, 30 or 29.5 days, how far has the tally
#    drifted from the Moon after 12 months, and after how many months is it a
#    whole day out?
# 2) If you alternate 29- and 30-day months, how does the error behave?

MONTH = 29.530589

def drift_days(months, tally_length):
    """How many days a tally of fixed month length has drifted from the Moon."""
    return months * (MONTH - tally_length)

for length in (29, 30, 29.5):
    print("tally", length, "-> drift after 12 months:", round(drift_days(12, length), 3), "days")

for length in (29, 30, 29.5):
    months_to_one_day = 1 / abs(MONTH - length)
    print("tally", length, "-> one day off after", round(months_to_one_day, 2), "months")

error = 0.0
length = 29
for month in range(1, 13):
    error += MONTH - length
    length = 30 if length == 29 else 29
    print(month, round(error, 3))

# Output when run:
# tally 29 -> drift after 12 months: 6.367 days
# tally 30 -> drift after 12 months: -5.633 days
# tally 29.5 -> drift after 12 months: 0.367 days
# tally 29 -> one day off after 1.88 months
# tally 30 -> one day off after 2.13 months
# tally 29.5 -> one day off after 32.69 months
# 1 0.531
# 2 0.061
# 3 0.592
# 4 0.122
# 5 0.653
# 6 0.184
# 7 0.714
# 8 0.245
# 9 0.775
# 10 0.306
# 11 0.836
# 12 0.367
#
# Reading the output: counting every month as 29 days lags the Moon by about
# 6.4 days in a year; counting 30 runs ahead by 5.6. A fixed 29.5 is far
# better: 0.37 days in 12 months, a whole day out after about 33 months. The
# alternating tally (29, 30, 29, 30...) is the same as using 29.5 on average:
# the error swings up and down by about half a day but creeps upward by
# 0.061 days per pair of months, ending at 0.367 days after 12 months.
# Real calendars therefore insert an occasional extra day. This is the same
# kind of fix that leap years make for the solar year.