# Need help with synodic months calculation

I’ve had an idea for a short story, but the plot hinges on a trick involving the difference between lunar and solar calendars. The idea is that someone will come into their inheritance when they are to the day 18 years old. The trick is that “18 years” is actually a mistranslation of a certain number of lunar months that’s just a bit shorter than 18 solar years. It would be either 222 or 223 synodic months, but given rounding errors I can’t tell which. Anyone care to tackle this?

Is the original figure from a lunar calendar like the Islamic one, or a solunar calendar like the Jewish or Chinese one? If the former, then you want 18 lunar years, which is 18*12 = 216 lunar months, putting you about a half-year short of the solar 18th birthday. If the latter, then it depends on when in the calendar cycle the person was born.

The easiest thing to do would be pick a culture (Jewish, Islamic, Chinese would be the best known to most people though there are others) and pick a set of years in which the story takes place and then go look at the appropriate calendar for those years.

The Islamic calendar has 12 lunar months, and is always shorter than the common year. The other two have 12 or 13 months and can be longer or shorter. There is an approximate 19 year matching cycle of lunar months and solar years so you have a bit of wiggle room to fit your 18 years in to get the result you want. There are several variations of the Chinese calendar.

Also I believe traditionally, the Chinese count your age from an imputed conception date. Someone relying on the traditional calendar might be doing that as well.

You can do it with synodic calendar rather easily…

A synodic month is the most familiar lunar cycle, defined as the time interval between two consecutive occurrences of a particular phase (such as new moon or full moon) as seen by an observer on Earth. The mean length of the synodic month is 29.53059 days (29 days, 12 hours, 44 minutes, 2.8 seconds). Due to the eccentricity of the lunar orbit around Earth (and to a lesser degree, the Earth’s elliptical orbit around the Sun), the length of a synodic month can vary by up to seven hours.

The 18 years could include 3,4 or 5 leap days.
Assume 4… Then
Add one if the first leap day occurs closer than two years after his birth, because then there is room for the 5 leap days with 16 years from start to finish.
Subtract one if one of the leap days would be in the century year that is NOT also divisible by 400.

So then sum up the days in those years… and consider what happens when you divide the period of 18*365.25 days (give or take a day ) into months of 29.53 days, rounded to the nearest day. That is. some months have 29 days, some have 30… alternating … 29 30 29 30 29 30 29 30 … with occassional doubling up … 29 30 30 29 30 29 30 29 30 29 30 …

Ok, I simplified things: I knew it would be either 222 or 223 synodic months, so I finally found a days-to-date calculator. 29.53059 days times 223 would be a bit over 6585 days, which is clearly longer than 18 years (around 6574 days, depending on the start and end dates). So 222 months it is. I’ll have to pick a date range for the 18 years to determine exactly how many days short that is, but that’s a minor point. Thanks all; this was one of those “I can only figure it out after I ask someone else” problems.

China, South Korea, and North Korea traditionally figure one’s age that way and popular usage is still the same; however, for all legal affairs, the current Gregorian calendar system and “actual age computed from birth” are used. The only legal affairs that the traditional Chinese calendar is used for are certain public holidays.