# Mathematics of the Jewish Calendar/Long-term data on the calendar

Since the calendar repeats exactly after 689,472 years, we can calculate the average properties of the calendar by working with data covering this time span. This may differ from the short-term results of analysing say a century of data.

## Number of years of each type

* Type 1: 39369
* Type 2: 81335
* Type 3: 43081
* Type 4: 124416
* Type 5: 22839
* Type 6: 29853
* Type 7: 94563
* Type 8: 40000
* Type 9: 32576
* Type 10: 36288
* Type 11: 26677
* Type 12: 45899
* Type 13: 40000
* Type 14: 32576
* Total: 689472
From this, we have for each possible weekday of:
Rosh Hashana:
* Mon: 193280
* Tue: 79369
* Thu: 219831
* Sat: 196992
Chanukah:
* Mon: 193280
* Wed: 151093
* Thu: 68738
* Fri: 69853
* Sat: 127139
* Sun: 79369
Fast of Tevet:
* Tue: 193280
* Wed: 26677
* Thu: 124416
* Fri: 138591
* Sun: 206508
New Year for Trees:
* Mon: 193280
* Tue: 26677
* Wed: 124416
* Thu: 138591
* Sat: 206508
And for each possible year length:
* 353 days: 69222
* 354 days: 167497
* 355 days: 198737
* 383 days: 106677
* 384 days: 36288
* 385 days: 111051

## Number of 19 year cycles

In the recurrence period, there are 36,288 19-year cycles. They may be tabulated by day of the week of the first day and by length.
* Mon: 9837
* Tue: 3811
* Thu: 12272
* Sat: 10368
* 6939 days: 17099
* 6940 days: 13648
* 6941 days: 5246
* 6942 days: 295
Note that fewer than 1% of cycles have 6942 days.

## Number of Molads

There are only 7 x 24 x 1080 = 181,440 possible Molads. Thus, over a full cycle, every Molad must occur three or four times. Molads with number of chalakim ending in 3 or 8 only occur three times; all the others occur four times. All 7 x 24 = 168 possible combinations of days and hours occur 4,104 times.
Two years with the same Molad Tishri, one ordinary and one leap, are of course always of different types. Two leap years with the same Molad Tishri are always of the same type. Two ordinary years with the same Molad Tishri are almost always of the same type; the only exception is where one year immediately follows a leap year and is subject to postponement rule 4 (Betuskapat), but the other year does not follow a leap year. For example, four years in the complete cycle have Molad 2d 15h 589ch: 88370, 205727, 396432 and 587137. 205727 is a leap year (type 9); 396432 is not postponed (type 2); the others are postponed (type 3).
The interval between two years with the same Molad is always either 117,357 or 190,705 years or the sum of these, 308,062 years. 689,472 = 3 x 190,705 + 117,357 = 2 x 190,705 + 308,062. If a molad occurs four times, then there are three gaps of 190,705 years and one of 117,357 years; if it occurs three times, then there are two gaps of 190,705 years and one of 308,062 years.
If the number of chalakim in the Molad has remainder 0, 1, 2, 3 and 4 respectively when divided by five, the number of ordinary years with that Molad are 2, 3, 2, 2 and 3; the number of leap years with that Molad are 2, 1, 2, 1 and 1.
Where there are three ordinary years per cycle with the same Molad, they are separated by gaps of 190705, 190705 and 308062 years. Where there are two years (ordinary or leap) with the same Molad, they are separated by gaps of 190705 and 498767 years. Where there is only one leap year with a given Molad, recurrence of course takes 689472 years.
Considering only the Molads of the first years of 19 year cycles, the number of chalakim always ends in 4 for odd cycles or 9 for even cycles; all possible Molads of these types occur exactly once over 689,472 years.
The first recurrence of the Molad for year 1 will be for year 117,358. It does not start a new cycle because it is the 14th, not the first, year of a 19 year cycle.

## Pairs of years of the same type

* It is impossible for two consecutive years to be of the same year type. This is easily seen: two consecutive years only begin on the same weekday if the first is an abundant leap year, and two consecutive years cannot both be leap years. (It is also impossible for pairs of leap years to be 4, 7, 12, 15, 18 years apart, or any interval a multiple of 19 greater than these.)
* It is also impossible for two years that are two, five, eight, 12, 15, 16, 22 or 25 years apart to be of the same type.
* Years three years apart can be of the same type, for all year types except 5. This is a rare occurrence for year types 1 and 6. For type 1 it would have happened in 3175/3178 if the present calendar had existed, and will next happen in 23130/23133. For type 6 it would have happened in 2990/2993 if the present calendar had existed, and will next happen in 55655/55658.
* Years four years apart can be of the same type, for all ordinary year types except 5; it is impossible for leap years, as pairs of leap years cannot be four years apart.
* Years six or 14 years apart can be of the same type, for year types 2, 4, 7 and 12. This is a rare occurrence for year type 12. It would have happened in 3173/3179 if the present calendar had existed, and will next happen in 35883/35889.
* Years seven or 20 years apart can be of the same type, for all ordinary year types (so seven is the smallest possible gap for year type 5); it is impossible for leap years, as pairs of leap years cannot be seven or 20 years apart.
* It is only possible for two years nine or 11 years apart to be of the same type if they are both of type 4.
* Years ten years apart can be of the same type, except for year types 1, 3, 5, 6 and 11.
* Years 13 and 18 years apart can be of the same type, for year types 4 and 7.
* Years 17 or 24 years apart can be of the same type, for all year types except 5.
* It is impossible for two years in the same position in consecutive 19 year cycles to be of the same type. The difference in Molad Tishri between such years is well over two days, wider than the Molad limits for any year type.
* Years 21 years apart can be of the same type, for year types 2, 4, 7, 8, 12 and 13.
* Years 23 years apart can be of the same type, for year types 2, 3, 4 and 7.
* Years 26 years apart can be of the same type, for year types 2, 4, 7.
* Years 27 years apart can be of the same type, for all year types; this is the smallest gap for which this is the case.
The longest possible gap between consecutive years of the same type is:
   * 1, 27 years
   * 2, 24 years
   * 3, 27 years
   * 4, 18 years
   * 5, 71 years
   * 6, 47 years
   * 7, 21 years
   * 8, 44 years
   * 9, 47 years
   * 10, 44 years
   * 11, 47 years
   * 12, 41 years
   * 13, 44 years
   * 14, 47 years
Gaps of 18 years between two consecutive type 4 years are rare; the first will be 42345/42363.
By far the longest possible gap is 71 years, for type 5; the last such gap was 5663/5734 and the next is 6255/6326.

## Possible year triplets

The following 52 sequences of year types in three consecutive years are possible:
* 1, 4, 9
* 1, 5, 10
* 1, 12, 4
* 2, 6, 10
* 2, 7, 11
* 2, 13, 4
* 2, 13, 5
* 2, 14, 6
* 3, 7, 11
* 3, 7, 12
* 3, 14, 6
* 3, 14, 7
* 4, 1, 12
* 4, 2, 13
* 4, 2, 14
* 4, 8, 7
* 4, 9, 1
* 4, 9, 2
* 5, 3, 14
* 5, 10, 2
* 6, 3, 14
* 6, 10, 2
* 7, 4, 8
* 7, 4, 9
* 7, 11, 3
* 7, 12, 4
* 8, 7, 4
* 8, 7, 11
* 8, 7, 12
* 9, 1, 4
* 9, 1, 5
* 9, 1, 12
* 9, 2, 6
* 9, 2, 13
* 10, 2, 6
* 10, 2, 7
* 10, 2, 13
* 10, 2, 14
* 11, 3, 7
* 11, 3, 14
* 12, 4, 1
* 12, 4, 2
* 12, 4, 8
* 12, 4, 9
* 13, 4, 2
* 13, 4, 9
* 13, 5, 3
* 13, 5, 10
* 14, 6, 3
* 14, 6, 10
* 14, 7, 4
* 14, 7, 11
Three consecutive years cannot all be ordinary; all sequences must be one of
* Ordinary, ordinary, leap
* Ordinary, leap, ordinary
* Leap, ordinary, ordinary
* Leap, ordinary, leap

## More on the 247 year recurrence

There are 24,073 times in a full cycle that there is a change in year type after a 247 year gap. It occurs 181 times for each year type in each possible position in the 19 year cycle; 24,073 = 181x7x19. (Each position in the cycle must be either always ordinary or always leap, so there are seven possible year types for each position.) Thus such a change occurs on average every 28 or 29 years. However, such changes are clustered. They only occur in 7,867 of the 19-year cycles and there may be up to seven changes in one cycle (e.g. cycles 233 and 436) though not six. Three consecutive years can have a change (e.g. 5521-3, 5933-5), but not four.
If there is only one change year in a 19 year cycle, it must be the 1st or 19th; often but not always, these form pairs, the 19th in one cycle and first in the next cycle. If there are two change years, they must be consecutive. If there are three, they must also be consecutive and not include the first year, unless they are the first, fourth and fifth year. If there are four, they form the following pattern: two consecutive, gap of two, two more consecutive. There is no simple pattern for five. For seven, they must be the 8th, 9th, 12th, 13th, 16th, 17th and 18th years of the cycle.
Put another way, if there are two change years in a 19 year cycle, any year may be involved, although year 1 is rare. If there are three, any year may be involved, although years 12 and 13 are rare. If there are four, any year but 17 may be involved, although years 9 and 13 are rare. If there are five, any year but 19 may be involved, although year 18 is rare.
The longest possible gap between "change years" is 183 years. The first such gap (had the calendar been in force then) was 3504-3687; the next is 7361-7544. Thus there is no full cycle of 247 years without a change. In fact, there will always be between two and 17 changes over any period of 247 years compared with the previous such period. The first period with 17 changes on the previous period is 11972-12218.