Mathematics of the Jewish Calendar/The Atbash
Once the day of the week of Rosh Hashana is known, so is the day of the week of every date from the previous 1st Adar (Adar Sheni in a leap year) until the next 29th Cheshvan. For dates before or after that, variations are possible depending on whether the previous year was a leap year and whether the current year is defective, regular or abundant (affecting the number of days in Cheshvan and Kislev).
A well-known mnemonic for calculating days of the week is the Calendar Atbash. An Atbash is a simple cypher where the first letter of the alphabet is replaced by the last, the second by the next to last, and so on. Thus Aleph is replaced by Tof, Beth by Shin and so on; this gives the acronym Atbash.
Applying the Atbash to the first seven days of Pesach, we get
- Aleph - Tof - Tisha B'Av
- Beth - Shin - Shavuot
- Gimel - Resh - Rosh Hashana
- Daled - Kuf - Keriat Hatorah, i.e. Simchat Torah, a day devoted to Keriat ("reading of") the Torah
- He - Tzadi - Yom Tzom Kippur, the Day of the Fast of Atonement
- Vav - Pe - Purim
- Zayin - Ayin - Yom ha-Atzmaut, Israel Independence Day
This is to be read "The first day of Pesach is on the same day of the week as the date beginning Tof, i.e. Tisha b'Av", etc. (The first line is spoilt if that day is Shabbat so that the fast has to be postponed to Sunday.) Israel Independence Day may also be moved. Note that the Atbash remained incomplete until the creation of the State of Israel meant that this new festival was created.
Since Rosh Hashana cannot fall on any of the three days Sunday, Wednesday or Friday, there are likewise three days of the week on which any given date from 1st Adar (Adar Sheni in a leap year) until 29th Cheshvan cannot fall. For dates before or after that, the situation is more complex; it is necessary to check the details of all fourteen possible types of year. Forbidden weekdays for some important dates are:
- Fast of Esther: Sunday, Tuesday; if it falls on Friday, it is observed on Thursday, instead, whereas if it falls on Saturday (Shabbat), it is postponed until Sunday. (When they fall on Friday or Shabat, all other fasts except Yom Kippur are observed the same. By design, Yom Kippur never falls on a Friday, and it is observed as a full fast when it falls on Shabbat.)
- Purim (14 Adar): Saturday, Monday, Wednesday (so Purim cannot fall on Shabbat except in places such as Jerusalem where it is observed a day late). Lag b'Omer is on the same day of the week as Purim.
- Pesach (1st day): Monday, Wednesday, Friday.
- Israel Independence Day (normally 5th Iyar, but movable): is subject to special rules, which have changed over the years; it can now only fall on Tuesday, Wednesday or Thursday.
- Shavuot (1st day): Tuesday, Thursday, Saturday. Next Hoshana Rabba is on the same day of the week as Shavuot.
- Fasts of Tammuz and Av: Monday, Wednesday, Friday; if they fall on Saturday (Shabbat) they are postponed to Sunday
- Rosh Hashana (1st day), 1st day Succot, Shemini Atzeret; Sunday, Wednesday, Friday.
- Fast of Gedaliah: Sunday, Wednesday, Friday, but if it falls on Saturday (Shabbat) it is postponed to Sunday
- Yom Kippur: Sunday, Tuesday, Friday.
- 1st day Chanukah: Tuesday.
- Fast of Tevet: Monday or Saturday (if so, the fast is postponed to Sunday); it can never be Wednesday in an ordinary year, or Thursday in a leap year.
- New Year for Trees: Sunday or Friday; it can never be Tuesday in an ordinary year, or Wednesday in a leap year.
- Purim Katan: Monday, Thursday, Saturday.