Mathematics of the Jewish Calendar/The four postponements of the New Year

The first day of Tishri, which is Rosh Hashana or New Year (literally, "Head of the Year"), should be on the day on which the molad falls, 12 months (or if the previous year was a leap year, 13 months) after the molad for the previous Rosh Hashana. However, more often than not the first day of Tishri is postponed by one or two days, following four rules known as the dechiyot (singular dechiyah).
Once the date of Rosh Hashana for that year and the following year have been calculated, we know how many days there need to be in that year. If there should be 354 or 384, the year is regular, so as noted above, so Cheshvan has 29 days and Kislev has 30. If there should be 355 or 385, Cheshvan has an extra day, and if there should be 353 or 383, Kislev loses a day.
Rule 1
If the time of the molad is after noon, Rosh Hashana is postponed to the next day.
Rule 2
If Rosh Hashana would fall on a Sunday, Wednesday or Friday, then it is postponed to the next day. If it has already been postponed to Sunday, Wednesday or Friday by Rule 1, it is thus postponed for two days.
The reason for this rule is that if Rosh Hashana fell on a Wednesday or Friday, then Yom Kippur (the Day of Atonement), which is on 10th Tishri, would be a Friday or Sunday. Since both Yom Kippur and the Sabbath, which occurs every Saturday, are days when virtually all work is forbidden, it would be very inconvenient to have them on consecutive days. However, it is all right for Yom Kippur and the Sabbath to coincide, as then they are only one day.
If Rosh Hashana fell on a Sunday, then Hoshana Rabba, which is 21st Tishri, would be a Saturday, the Sabbath. It is traditional on that day to walk seven times round the Synagogue carrying a palm branch; this ritual could not be performed on the Sabbath.
This rule is often called Lo ADU Rosh. "Lo" is the Hebrew for "no". "ADU" represents the first, fourth and sixth letters of the Hebrew alphabet, and means that Rosh (Hashana) cannot fall on the first, fourth or sixth days of the week.
If these were the only postponement rules, it would be possible for an ordinary year to have 356 days or a leap year to have 382 days. To avoid this, there are two further rules.
Rule 3
If the calculated day of the New Moon is Tuesday, the calculated time is at least 9 hours 204 chalakim, and the year will be ordinary, the New Year is postponed; it cannot be on Wednesday by Rule 2 so it is moved to Thursday.
Molad Tishri of the following year will fall on Saturday at or after 18hr (noon), so by rules 1 and 2 the next Rosh Hashana would be postponed to Monday. Without this rule, the year would then have 356 days. This rule ensures that it has only 354.
A year postponed by this rule always becomes type 4, and the previous year becomes type 7 or 12, depending on whether it is an ordinary or a leap year.
Rule 4
If the calculated day of the New Moon is Monday, the calculated time is at least 15 hours 589 chalakim, and the year will be the year after a leap year, the New Year is postponed to Tuesday.
Molad Tishri of the previous leap year fell on or after Tuesday at 18hr (noon), so by rules 1 and 2 the previous Rosh Hashana was postponed to Thursday. Without this rule, that year would then have had only 382 days. This rule ensures that it has 383.
A year postponed by this rule always becomes type 3, and the previous year becomes type 11.
How frequently is Rosh Hashana postponed?
If the Molad of Tishri falls on any of three days of the week, i.e. Sunday, Wednesday or Friday, Rosh Hashana is always postponed. Clearly, this happens on average in 3/7 of all years or about 43%.
If the Molad of Tishri falls on any of the other four days of the week, Rosh Hashana is only postponed if the Molad falls in the last quarter of the day. This happens on average in (4/7)/4 = 1/7 of all years or about 14%. Thus in total, from the first two rules, it is postponed in 4/7 or about 57% of all years.
If the Molad falls after noon on Saturday, Tuesday or Thursday, Rosh Hashana is postponed by two days. This happens on average in (3/7)/4 = 3/28 of all years or about 11%.
Rules 3 and 4 take effect more rarely. Rule 3 only affects ordinary years, i.e. 12 in a 19 year cycle. The range of Molads affected is 8 hours 876 chalakim or 5.24% of a week, so on average it applies to about 12/19 x 5.24% or 3.31% of years. Rule 4 only affects ordinary years that follow a leap year, i.e. 7 in a 19 year cycle. The range of Molads affected is 2 hours 491 chalakim or 1.46% of a week, so on average it applies to about 7/19 x 1.46% or 0.54% of years. (The last such year was 5766 and the next is 6013.) Thus only 3.85% of Rosh Hashanas are postponed for these reasons, so in total 61.0% of Rosh Hashanas are postponed.
Leap years are postponed less often than ordinary years, since rules 3 and 4 do not apply. Thus they are only postponed 57.1% of the time. Years immediately following leap years, where all rules apply, are postponed 63.8% of the time; other years, where rule 4 does not apply, are postponed 62.4% of the time.
The "Four Gates".
Saadia Gaon (10th century CE) refers to the "Arbaah Shaarim" ("Four gates") as an ancient rule for regulating the calendar. It can be expressed as a table that shows, for each of four possible categories of years:
the range of Molads that correspond with each possible year type. The rule is exactly equivalent to the four dechiyyot.