In honor of Rosh Chodesh Tevet. (You'll see why!)
Every Hebrew calendar geek knows the "ATBaSh" parlor trick, where if you know the day of the week of (almost) any Jewish holiday, you can quickly figure out the day of the week of (almost) any other holiday that year. As we have blogged before, this works only for the period from Adar through Cheshvan. Fortunately, that period includes all of the major holidays, and a few minor ones too. But it doesn't cover the minor holidays that fall during the winter, and that's what this post will seek to do.
The trick dates back at least to the Tur (14th century) and it works like this: Take the first six days of Pesach in a given year (note that the period from Adar to Cheshvan spans two Hebrew years, so we're looking at a given Gregorian year), and write the Hebrew alphabet backwards, starting from the end. You'll find that the corresponding holidays fall on the same day of the week as that day of Pesach.
1) תשעה באב = ת Tish'ah B'Av is always exactly 16 weeks after the first day of Pesach. (Also, 17 Tammuz is 3 weeks before 9 Av, and therefore the same day of the week. When they fall on Shabbat, as they will in 2012, the actual observance is delayed to Sunday.)
2) שבועות = ש Shavuot, by definition, is 7 weeks after the second day of Pesach.
3) ראש השנה = ר Rosh Hashanah. (Sukkot and Shemini Atzeret are also on the same day of the week.)
4) קריאת התורה = ק In communities that observe two days of Shemini Atzeret, this is "Simchat Torah" (on the second day of Shemini Atzeret).
5) צום כפור = צ Yom Kippur (9 days after Rosh Hashanah, and therefore 2 days of the week later). (Also, Tzom Gedaliah is 1 week before Yom Kippur and thus the same day of the week, except when it is delayed due to Shabbat.)
6) פורים = פ Purim (in unwalled cities).
The original version just covered the first six days of Pesach, but the 7th day was added in the 20th century:
7) עצמאות = ע Israeli Independence Day (at least before the Knesset starts mucking with the date).
It works so perfectly that one wonders whether this was the real reason that Ben-Gurion decided to declare independence a day before the British Mandate expired.
Other Israeli civil observances tied to the Hebrew calendar can also be located with this framework. Yom Hashoah is always the same day of the week as Purim (again, before the Knesset reschedules it); to remember this, note that some have suggested that Yom Hashoah is the Purim story without Esther. Yom Yerushalayim is exactly one week before Shavuot.
Finally, Lag Ba'Omer is also the same day of the week as Purim: not hard to remember.
Rosh Hashanah can fall on only four days of the week (Monday, Tuesday, Thursday, Shabbat), and therefore all of these other days are also restricted to four days. To figure out which four days, just use the relationships above. For example, Shavuot is on the day (of the week) before Rosh Hashanah, so it can only fall on Sunday, Monday, Wednesday, or Friday.
During the winter months, it's not so simple. This is because there are three variables that can cause the calendar to be different from one year to the next:
1) Cheshvan can have 29 or 30 days.
2) Kislev can have 29 or 30 days.
3) There can be one or two months of Adar. (In leap years, Adar I is the "extra" month, and always has 30 days.)
In order to connect winter holidays to non-winter holidays, you need to know at least one of those three pieces of information. To keep it as simple as possible, the mnemonics below will be for a year that goes from Tevet to Kislev. (As a convenient and coincidental memory aid, this corresponds roughly to the Gregorian year, but not precisely: 10 Tevet can fall in either December or January. Thus some Gregorian years have two Fasts of Tevet, and some have none.) This way, you only have to know one additional variable: the number of days in Cheshvan (to expand forward into Kislev), or whether it's a leap year (to expand backward into Tevet and Sh'vat).
Let's start with days that depend only on whether it's a leap year, since that's something you're more likely to know off the top of your head.
Tu BiShvat: In a leap year, it's on the same day of the week as Rosh Hashanah. (Remember, this is the following Rosh Hashanah.) In a non-leap year, it's on the same day as Yom Kippur. Mnemonic: New Year of the Trees.
Since Tu BiShvat is the same day as Rosh Hashanah or Yom Kippur, there are five possible days of the week when it can fall: Monday, Tuesday, Wednesday, Thursday, or Shabbat. (Neither Rosh Hashanah nor Yom Kippur can fall on Friday or Sunday.)
10 Tevet: It's always one day (of the week) after Tu BiShvat. Mnemonic: Deuteronomy 20:19 says that when you besiege a city, you shouldn't cut down the trees. Trees before siege.
Thus, 10 Tevet can also fall on five days of the week: Sunday, Tuesday, Wednesday, Thursday, or Friday. As a guest post here discussed, this makes it the only fast day that can fall on a Friday. As discussed there also, it can never fall on Shabbat, which makes the question of whether we would still fast purely hypothetical.
Finally, Chanukah. It's complicated because whether Cheshvan has 29 or 30 days in a given year isn't something we're likely to know without looking up. So here's a quick way to find that from information you're more likely to have handy (which just happens to also be the way it's determined in the calendar algorithm itself).
You need to know the day of the week of Rosh Hashanah this year and next year, and whether it's a leap year. From the number of days in between, you can figure out how many days are in the year. Non-leap years have 353, 354, or 355 days, and leap years have 383, 384, or 385 days, and it helps to remember that 350 and 385 are both divisible by 7. If the year has 353, 354, 383, or 384 days, then Cheshvan has 29 days; if the year has 355 or 385 days, then Cheshvan has 30 days.
Once you know that, then you can find which day of the week Chanukah begins on. If Cheshvan has 30 days, then Chanukah begins on the same day of the week as Rosh Hashanah (exactly 12 weeks later). If Cheshvan has 29 days, then Chanukah begins one day earlier, on the same day of the week as Shavuot. Mnemonic: Applesauce or sour cream?
Since Chanukah can begin on the same day as Rosh Hashanah or a day earlier, there are six days of the week when it can begin: all of them except Tuesday, because years beginning on Tuesday can only have 354 or 384 days, so in those years, Cheshvan always has 29 days (so Chanukah begins on Monday).
To see this in action, let's use this year (5772) as an example.
Starting with Chanukah: it's in 2011, so it goes with that set of holidays (Pesach on Tuesday, Rosh Hashanah on Thursday, etc.). Did Cheshvan have 29 or 30 days? Well, I know that Rosh Hashanah this year was on Thursday, and next year it's on a Monday, and this isn't a leap year. From Thursday to Monday is a 4-day gap, so this year must have 350+4 = 354 days. That means Cheshvan had 29 days. So Chanukah began on the same day of the week as Shavuot (one day before Rosh Hashanah): Wednesday.
For the minor holidays in Tevet and Sh'vat, we look instead at the upcoming holidays in 2012 (when Rosh Hashanah is on a Monday). It's not a leap year, so Tu BiShvat is on the same day as Yom Kippur: Wednesday. 10 Tevet is one day later, on Thursday.