# Mathematics of the Jewish Calendar/The fourteen types of year

There are fourteen possible types of year. Two years are of the same type if they have the same number of days and every date in one year falls on the same weekday in the other year.
As explained above, a year may not start on Sunday, Wednesday or Friday. Thus it may start on any of four weekdays (Monday, Tuesday, Thursday or Saturday). It may be an ordinary year or a leap year, and it may be defective, regular or abundant. Thus there can at most be 4 x 2 x 3 = 24 year types.
Of these, nine are obviously impossible because they would cause the next Rosh Hashana to fall on a forbidden weekday. For example, if a leap year starts on Monday, it can be deficient or abundant, because then the next Rosh Hashana would fall on Saturday or Monday respectively, but it cannot be regular, because then the next Rosh Hashana would fall on Sunday.
Considering the possible Molad limits, one possible type of leap year (Rosh Hashana Tuesday, abundant) can never happen.
This leaves fourteen possible year types, and they all do occur.

## Nomenclature of year types

There are two systems used to denote year types. In the first, the weekday of Rosh Hashana is given by an integer based on its position in the week. Thus, Monday is 2, being the second day of the week, and similarly Tuesday, Thursday and Saturday are respectively 3, 5 and 7. There is then a letter to denote the year's length, lower case (d, r, a for deficient, regular and abundant) for ordinary years and capital (D, R, A) for leap years.
In the second, there are again a numeral and a letter, and then there is another digit giving the weekday of the first day of Pesach. This second digit enables a distinction between ordinary and leap years, so it does not matter whether the letter is a capital or lower case.
The fourteen types, in the two systems, are:
* 1 = 2d = 2d3
* 2 = 2a = 2a5
* 3 = 3r = 3r5
* 4 = 5r = 5r7
* 5 = 5a = 5a1
* 6 = 7d = 7d1
* 7 = 7a = 7a3
* 8 = 2D = 2D5
* 9 = 2A = 2A7
* 10 = 3R = 3R7
* 11 = 5D = 5D1
* 12 = 5A = 5A3
* 13 = 7D = 7D3
* 14 = 7A = 7A5
It follows that:
* A year beginning on Tuesday is always regular, whether it is an ordinary or leap year.
* A year beginning on Monday or Saturday is never regular, whether it is an ordinary or leap year.
* A year beginning on Thursday is never deficient if it is an ordinary year, and never regular, if it is a leap year.

## Possible sequences of two consecutive years

When considering which sequences of two consecutive years are possible, note that two consecutive years cannot both be leap years. Days of the week must be consistent: Rosh Hashana in the second year must always be two weekdays after Pesach in the first year. Thus in year type 1, Pesach is on Tuesday so the next year must be one where Rosh Hashana is Thursday.
To determine which pairs can actually occur, it is necessary to calculate the range of possible Molads that will produce the first year type. Then add on the shift in Molad due to an ordinary year or a leap year, as the case may be, and see what types of year are produced by the resultant range of possible Molads.
Thus only the following sequences can occur.
* Type 1 may be followed by types 4,5,12
* Type 2 may be followed by types 6,7,13,14
* Type 3 may be followed by types 7,14
* Type 4 may be followed by types 1,2,8,9
* Type 5 may be followed by types 3,10
* Type 6 may be followed by types 3,10
* Type 7 may be followed by types 4,11,12
* Type 8 may be followed by type 7
* Type 9 may be followed by types 1,2
* Type 10 may be followed by type 2
* Type 11 may be followed by type 3
* Type 12 may be followed by type 4
* Type 13 may be followed by types 4,5
* Type 14 may be followed by types 6,7
Two consecutive years cannot both be regular or both deficient, but they can both be abundant. Regular leap years are always followed by abundant years. Regular common years that begin on Tuesday are always followed by abundant common or leap years.