Turning 27 or 30 in the 19 year cycle - some math
Note: This is taken from a Facebook post on 7/20/21
A lot of people are familiar with the 19 year cycle where Hebrew and English birthdays (as they are commonly called and I will call them for simplicity sake) are usually on the same day, but people often notice that when they turn 27 or 30 their birthdays can be within a day of each other. I've been getting asked about this more and more (including today) and thought I'd explain (as simply as possible):
A month in the Jewish calendar is determined by the moon's rotation, with the beginning of the month being a "new moon" (I don't know the exact science well enough so I won't get into it), with each so called lunar cycle being approximately 29 1/2 days long. A "normal" year consists of 12 months, averaging 29 1/2 days by alternating months of 30 and 29 days, totaling 354 days (though a year can also be 353 or 355 days). A leap year adds an additional 30 day month (Adar 1), making a "normal" leap year 384 days (most leap years are actually 383 or 385, but let's use 384 for simplicity).
Using the above numbers and assuming a 365 day solar year (for simplicity), a non-leap Hebrew year is 11 days (365-354) shorter than an English year, so a Hebrew date will get 11 days earlier the following year. For the easiest example of this (which conveniently applies to me and someone I know a year older), if your Hebrew birthday falls out on December 31st one year, it will be December 20 the next as the Hebrew year is 11 days shorter). By contrast, a leap Hebrew year is 19 days (384-365) longer than an English year, so a Hebrew date will get 19 days later.
The 19 year cycle consists of 7 leap years and 12 non-leap years. The 7 leap years (birthday getting 19 days later each time) account for 7*19=133 gained Hebrew days, while the 12 non-leap years (birthday getting 11 days earlier) account for 12*11=132 lost Hebrew days, essentially balancing out.
It's worth noting that the leap years are actually very specific, occurring almost once every 3 years with 2 exceptions. In a full cycle, leap years occur on years 3/6/8/11/14/17/19. For reference, the most recent cycle began on Rosh Hashanah 2016.
When someone turns 27 or 30, they're either 8 or 11 years into their second 19 year cycle. Using the leap year occurrences above, an 8 year cycle will almost always* consist of 3 leap years and 5 non-leap years, while an 11 year cycle will almost always* consist of 4 leap years and 7 non-leap years. Using similar math, the 8 year cycle gains 3*19=57 Hebrew days while losing 5*11=55 Hebrew days, almost balancing out. The 11 year cycle gains 4*19=76 Hebrew days while losing 7*11=77 Hebrew days, similarly almost balancing out. That is why on those years birthdays can be a day apart (my 27th Hebrew birthday began the night of my English birthday).
*The one exception for the 8 year cycle is year 8/9 of the cycle (if born between Adar 2 of year 8 and Adar of year 9), where only 2 leap years are encountered (years 11/14) in a span of 8 years. In such a case, you only gain 2*19=38 Hebrew days, but lose 6*11=66 Hebrew days, resulting in a 28 days loss. For reference, my Hebrew birthday (29 Kislev, 5th day of Chanukah) was December 30th in 2005 (year 9 of the cycle that began on RH 1997) and December 2nd in 2013 (year 17, but before the leap month occurred).
The exception for the 11 year cycle is the reverse, with 5 leap years (17/19/3/6/8) in that span to go with only 6 non-leap years,. This results in gaining 5*19=95 Hebrew days but losing only 6*11=66 Hebrew days, resulting in a 29 day gain. As expected, my Hebrew birthday will be December 30th again in 2024, 28 days later than the aforementioned 2013 date.
Lastly, as mentioned before, most of the above is simplified. There are obviously other factors that vary from year to year and the numbers don't always work out perfectly. Notably, my 30th Hebrew birthday was 3 days before my English one, but someone who turned 19 that year will actually have their birthdays align perfectly at age 30 (or if they're born that year, at age 11). More often than not though, you don't need to turn a multiple of 19 to have Hebrew and English birthdays very close to each other.