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Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle.

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Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.

This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.

These three phenomena are independent of each other, so there is no direct correlation between them.

On average, the moon revolves around the Earth in about 29½ days.

After many years of blissful ignorance, I finally sat down and worked out the mathematics involved, and I have added a page on The Jewish Calendar: A Closer Look, which may be of interest to those who want a deeper understanding or who want to write a Jewish calendar computer program.