Within the lunar year, there are 364 days, so in order to not fall behind, ever third year we add an intercalary month that consists of 30 days. That is the yellow segment you see in the picture. This little cycle happens again - 3 years, at the end of the third year adding an intercalary month. But then, in order to completely align ourselves to the solar cycle, what we do is: after two years, add an intercalary month. (FYI, FTR, that little yellow segment, the intercalary month, is part of the third year, not the fourth)
Now, in the picture above, you can see the little cycle. For reference, the lunar year has 12 months. Month One will have 30 days, month two 29, month three 30, etc. The third year, however, will have 13 months (because of the intercalary month).
Now let's say that the first day ($x$, shown in the picture) of the first 3 year cycle is Sunday. After the intercalary month, year four, $x$ = Sunday, also for the beginning of the 6th year. However, once the two year cycle ends, 9th year begins (a new lunar cycle), the first day, $x$, is now Wednesday.
From this, I assumed that the next 3 year cycle, and the beginning of the 4th year, will be Wednesday.
From this you can also assume, that the first day of the third lunar cycle will be $x + 3 = Saturday$
How can I use this information to derive a formula that will let me calculate $x$ for the a given lunar cycle (i.e, the day of the week that the lunar cycle will begin on? (e.g. Lunar cycle 189 will start on Saturday....)
I am appealing to help on this site, because it is the only place I know of (online) where a bunch of people are willing to give up their time to answer questions - so, thank you a ton!
I haven't got the slightest idea of how to solve this, or even if it's possible (however, I think it is highly probable that it is possible).
Thanks a ton, guys!