The First Day of the Month
The First Day of the Month

by Remy Landau
=================================

The full Hebrew calendar cycle of 689,472 years consists of 8,527,680 months. This total can be easily derived by multiplying the number of months in every 19 year cycle with the number of these cycles in the full Hebrew calendar cycle. This gives 235 months * 36,288 = 8,527,680 months.

Each day of the week will coincide with the first day of a month. However, due to the initial Rosh Hashannah rule of Lo ADU it is not possible for some of the months to begin on certain days of the week. For example, Tishrei cannot begin on either Sunday, Wednesday, or Friday. Similarly, Kislev never begins on a Saturday, while the month of Tevet avoids starting on either Thursday or Saturday.

In here, the 30 day month of Adar is called Leap Adar and the weekday distribution of its first day is appended to the folowing table.

The following table demonstrates the weekday distribution of the first day of each month over the full Hebrew calendar cycle.

Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 193280 79369 0 219831 0 196992 689472
Heshvan 0 196992 0 193280 79369 0 219831 689472
Kislev 151093 68738 69853 127139 79369 193280 0 689472
Tevet 193280 26677 124416 138591 0 206508 0 689472
Shevat 0 193280 26677 124416 138591 0 206508 689472
Adar 0 206508 0 193280 26677 124416 138591 689472
v'Adar 0 85899 0 72576 0 68864 26677 254016
Nisan 79369 0 219831 0 196992 0 193280 689472
Iyar 0 193280 79369 0 219831 0 196992 689472
Sivan 196992 0 193280 79369 0 219831 0 689472
Tammuz 219831 0 196992 0 193280 79369 0 689472
Av 0 219831 0 196992 0 193280 79369 689472
Elul 193280 79369 0 219831 0 196992 0 689472
Totals 1033845 1463854 989787 1345474 1153940 1282540 1258240 8527680
Leap Adar 0 72576 0 68864 26677 0 85899 254016

Some Observations on the First Days

The months of Iyar and Tishrei have exactly the same weekday distribution for their first days. This is because the first day of the month of Iyar is exactly 147 days away from the first day of the subsequent Tishrei. Since 147 days is exactly 21 weeks, then whichever day of the week begins Iyar, will also begin Tishrei. For example, whenever the first day of Iyar is Monday, the first day of the following Rosh Hashannah will also be Monday, as happens to be the case for Iyar 5758H (1998g) and Rosh Hashannah 5759H (1998g).

The month of Kislev can begin on every single day of the week except for Saturday. This will be true for both the non-leap and the leap year distibutions.

The months of Kislev, Tevet, Sivan, Tammuz, and Elul never begin on Saturday.

The most likely start day for any month is Monday. From the above table it can be seen that the first days of the months coincide some 17.2% of the time with Monday.

Similarly, the least likely start day for any month is Tuesday. From the above table it can be seen that the first days of the months coincide some 11.6% of the time with Tuesday.

The weekday distribution of the first days for the months of Heshvan, Nisan, Iyar, Sivan, Tammuz, Av, and Elul have the same values as those for the month of Tishrei, although each may be shifted differently across the weekdays. That is because the first days of each of these months is a constant distance away from the first day of the month of Tishrei.

The month of Adar has 3 different characteristics. The first is the 29 day month which occurs in all non-leap years. The second is the 30 day month of Adar which is inserted after the month of Shevat during Hebrew leap years. And the third is the 29 day month of v'Adar which always follows the leap month of Adar.

It is to be noted that v'Adar has the same weekday distribution values as those for the month of Leap Adar. However, the values for v'Adar are shifted two days later than those for Leap Adar.

Adar cannot start on Friday in a leap year, and can never begin either on Sunday or Tuesday.

The First Day Distributions by Keviyyot

The following tables show the week day distribution of the first day of the months segregated by year lengths.

It is interesting to note that for the following year lengths, these months have exactly the same start day distribution as the month of Tishrei

```          353 days - Adar   Elul
354 days - Tammuz
355 days - Shevat
384 days - Sivan
385 days - Shevat Iyar
```

It is also interesting to note that the most likely week day for the first day of a month varies by year length as follows

```          353 days - Monday
354 days - Sunday Tuesday Thursday Friday (each occurs 334,994 times)
355 days - Monday
383 days - Monday
384 days - Friday
385 days - Monday
```

The First Day Distribution Surprise

Each year type occurs with a particular frequency over the full Hebrew calendar cycle of 689,472 years. The big surprise is that over this cycle some of the week days actually catch the first days of the month with exactly the same frequency as the year types.

Week Day Firsts of Same Frequency as Year Types
Year LengthOccurencesWeek Day
353
`69,222`
Sunday
354
`167,497`
Monday
355
`198,737`
Tuesday
383
`106,677`
NONE
384
`36,288`
Wed & Sat
385
`111,051`
Tuesday

The 383 day years are unique in that none of its week days catch the first day of a month with the same frequency of occurrence as the 383 day years.

Some of the week days also catch the first days of the month with exactly double the frequency of the year types.

Week Day Firsts of Double Frequency as Year Types
Year LengthOccurencesWeek Day
353
`2*69,222`
Wednesday
354
`2*167,497`
Tu & Th & Fr
355
`2*198,737`
NONE
383
`2*106,677`
Thursday
384
`2*36,288`
Su & Mo & Tu & Th
385
`2*111,051`
Saturday

The 355 day years are unique in that none of its week days catch the first day of a month with double the frequency of occurrence as the 355 day years.

The 384 day years are the only ones to have a week day, namely Friday, catch the first day of a month at triple the frequency of occurence of the 384 day years.

For 353 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 39369 0 0 0 0 29853 69222
Heshvan 0 29853 0 39369 0 0 0 69222
Kislev 0 0 29853 0 39369 0 0 69222
Tevet 0 0 0 29853 0 39369 0 69222
Shevat 0 0 0 0 29853 0 39369 69222
Adar 0 39369 0 0 0 0 29853 69222
v'Adar 0 0 0 0 0 0 0 0
Nisan 29853 0 39369 0 0 0 0 69222
Iyar 0 0 29853 0 39369 0 0 69222
Sivan 0 0 0 29853 0 39369 0 69222
Tammuz 39369 0 0 0 0 29853 0 69222
Av 0 39369 0 0 0 0 29853 69222
Elul 0 29853 0 39369 0 0 0 69222
Totals 69222 177813 99075 138444 108591 108591 128928 830664

For 354 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 0 43081 0 124416 0 0 167497
Heshvan 0 0 0 0 43081 0 124416 167497
Kislev 124416 0 0 0 0 43081 0 167497
Tevet 43081 0 124416 0 0 0 0 167497
Shevat 0 43081 0 124416 0 0 0 167497
Adar 0 0 0 43081 0 124416 0 167497
v'Adar 0 0 0 0 0 0 0 0
Nisan 0 0 0 0 43081 0 124416 167497
Iyar 0 124416 0 0 0 0 43081 167497
Sivan 43081 0 124416 0 0 0 0 167497
Tammuz 0 0 43081 0 124416 0 0 167497
Av 0 0 0 43081 0 124416 0 167497
Elul 124416 0 0 0 0 43081 0 167497
Totals 334994 167497 334994 210578 334994 334994 291913 2009964

For 355 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 81335 0 0 22839 0 94563 198737
Heshvan 0 94563 0 81335 0 0 22839 198737
Kislev 0 22839 0 94563 0 81335 0 198737
Tevet 81335 0 0 22839 0 94563 0 198737
Shevat 0 81335 0 0 22839 0 94563 198737
Adar 0 94563 0 81335 0 0 22839 198737
v'Adar 0 0 0 0 0 0 0 0
Nisan 22839 0 94563 0 81335 0 0 198737
Iyar 0 0 22839 0 94563 0 81335 198737
Sivan 81335 0 0 22839 0 94563 0 198737
Tammuz 94563 0 81335 0 0 22839 0 198737
Av 0 94563 0 81335 0 0 22839 198737
Elul 0 22839 0 94563 0 81335 0 198737
Totals 280072 492037 198737 478809 221576 374635 338978 2384844

For 383 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 40000 0 0 26677 0 40000 106677
Heshvan 0 40000 0 40000 0 0 26677 106677
Kislev 26677 0 40000 0 40000 0 0 106677
Tevet 0 26677 0 40000 0 40000 0 106677
Shevat 0 0 26677 0 40000 0 40000 106677
Adar 0 40000 0 0 26677 0 40000 106677
v'Adar 0 40000 0 40000 0 0 26677 106677
Nisan 26677 0 40000 0 40000 0 0 106677
Iyar 0 0 26677 0 40000 0 40000 106677
Sivan 40000 0 0 26677 0 40000 0 106677
Tammuz 40000 0 40000 0 0 26677 0 106677
Av 0 40000 0 40000 0 0 26677 106677
Elul 0 26677 0 40000 0 40000 0 106677
Totals 133354 253354 173354 226677 213354 146677 240031 1386801

For 384 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 0 36288 0 0 0 0 36288
Heshvan 0 0 0 0 36288 0 0 36288
Kislev 0 0 0 0 0 36288 0 36288
Tevet 36288 0 0 0 0 0 0 36288
Shevat 0 36288 0 0 0 0 0 36288
Adar 0 0 0 36288 0 0 0 36288
v'Adar 0 0 0 0 0 36288 0 36288
Nisan 0 0 0 0 0 0 36288 36288
Iyar 0 36288 0 0 0 0 0 36288
Sivan 0 0 36288 0 0 0 0 36288
Tammuz 0 0 0 0 36288 0 0 36288
Av 0 0 0 0 0 36288 0 36288
Elul 36288 0 0 0 0 0 0 36288
Totals 72576 72576 72576 36288 72576 108864 36288 471744

For 385 Day Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 32576 0 0 45899 0 32576 111051
Heshvan 0 32576 0 32576 0 0 45899 111051
Kislev 0 45899 0 32576 0 32576 0 111051
Tevet 32576 0 0 45899 0 32576 0 111051
Shevat 0 32576 0 0 45899 0 32576 111051
Adar 0 32576 0 32576 0 0 45899 111051
v'Adar 0 45899 0 32576 0 32576 0 111051
Nisan 0 0 45899 0 32576 0 32576 111051
Iyar 0 32576 0 0 45899 0 32576 111051
Sivan 32576 0 32576 0 0 45899 0 111051
Tammuz 45899 0 32576 0 32576 0 0 111051
Av 0 45899 0 32576 0 32576 0 111051
Elul 32576 0 0 45899 0 32576 0 111051
Totals 143627 300577 111051 254678 202849 208779 222102 1443663

First Day Distribution Summary By Year Type

Summary For NON-LEAP Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 120704 43081 0 147255 0 124416 435456
Heshvan 0 124416 0 120704 43081 0 147255 435456
Kislev 124416 22839 29853 94563 39369 124416 0 435456
Tevet 124416 0 124416 52692 0 133932 0 435456
Shevat 0 124416 0 124416 52692 0 133932 435456
Adar 0 133932 0 124416 0 124416 52692 435456
v'Adar 0 0 0 0 0 0 0 0
Nisan 52692 0 133932 0 124416 0 124416 435456
Iyar 0 124416 52692 0 133932 0 124416 435456
Sivan 124416 0 124416 52692 0 133932 0 435456
Tammuz 133932 0 124416 0 124416 52692 0 435456
Av 0 133932 0 124416 0 124416 52692 435456
Elul 124416 52692 0 133932 0 124416 0 435456
Totals 684288 837347 632806 827831 665161 818220 759819 5225472

Summary For LEAP Years
Start of Month Distribution by Week Day
Sun Mon Tue Wed Thu Fri Sat Totals
Tishrei 0 72576 36288 0 72576 0 72576 254016
Heshvan 0 72576 0 72576 36288 0 72576 254016
Kislev 26677 45899 40000 32576 40000 68864 0 254016
Tevet 68864 26677 0 85899 0 72576 0 254016
Shevat 0 68864 26677 0 85899 0 72576 254016
Adar 0 72576 0 68864 26677 0 85899 254016
v'Adar 0 85899 0 72576 0 68864 26677 254016
Nisan 26677 0 85899 0 72576 0 68864 254016
Iyar 0 68864 26677 0 85899 0 72576 254016
Sivan 72576 0 68864 26677 0 85899 0 254016
Tammuz 85899 0 72576 0 68864 26677 0 254016
Av 0 85899 0 72576 0 68864 26677 254016
Elul 68864 26677 0 85899 0 72576 0 254016
Totals 349557 626507 356981 517643 488779 464320 498421 3302208

```First  Paged  1 Jul 1997