### Now's The Time(s) To Say "I'm Sorry"

But how many times - there's the rub! With that, here are three "Torah math" quiz questions relating to this time of year. Two are new, the third a golden oldie from the early days of this blog, also question 3 in the above link.

1) This week we read Parshas Nitzavim by itself, and then Vayelech by itself next Shabbos. Many other years, they are read on a single Shabbos as a double parshah. About what percentage of the time are they read separately vs. together? (Note: working out the answer to this item should help you somewhat with the next one.)

2) Under Ashkenazic minhag, we start reciting selichos on the Sunday morning before Rosh Hashanah - or sometimes two Sundays before as this year - and then conclude on erev Yom Kippur. So, not counting the selichos said in maariv and neilah of Yom Kippur itself, what is the average number of days each year on which selichos for yomim noraim are said?

3) From mincha on erev Yom Kippur through Neilah, inclusive, how many times is the chest struck? Note: again, this is based on the Ashkenazic minhag/siddur, and assumes the chest is not struck for the "al chataim" lines at the end of the "al chayt" confessional.

## 22 Comments:

1) If Nitzavim and Vayelech are read individually during leap years and as a double parshah during non-leap years, then during the nineteen-year cycle they are read separately seven times and together twelve times, making a percentage of just about 37% -- the actual number has a lovely repeating decimal -- of the time they are read separately compared to together.

But is the schedule based on leap years, or on leap years alone? Some doubling has to do with when a holiday falls, so that there is a holiday Torah reading instead of the weekly one. But since you don't specify outside of Israel as the context for this question, I am going to guess the answer is indeed simply based on leap years.

I don't have enough time before candlelighting to work on the rest, so they will have to wait for next week.

Yay for Jewish math questions! Thanks Elie!

O Sailor, Come Ashore

(Part I)

O sailor, come ashore

What have you brought for me?

Red coral , white coral,

Coral from the sea.

(Part II)

I did not dig it from the ground ，

Nor pluck it from a tree;

Feeble insects made it

In the stormy sea.

～by rs gold

So I've been thinking about this. There's never any difference in the schedule of Torah readings inside and outside of Israel at this time of year anyway, but that doesn't mean the answer doesn't have to do with when during the week Rosh Hashanah falls, with how many regular weekly Torah readings can fit in around all the holidays. Now, the cycle of when during the week a holiday falls is a cycle much longer than the nineteen-year cycle, many times longer, in fact, than the years we have so far counted. So if the answer is based only on the leap years, then the answer is the 37%, but, more likely, if it involves when during the week Rosh Hashanah falls, well, I simply don't know how to begin. I still can't figure out why Bamidbar didn't immediately precede Shavuos this past year!

Answering the second question certainly necessitates knowledge of how often Rosh Hashanah falls on which day of the week. As for with the first question, I don't know where to begin with that. Am I making all this much more complicated than it really is?

I ended up reading the earlier answer to the Golden Oldie that is the third question. My knowledge of television seems to be stronger than my knowledge of tefilah. I do so wish it were the other way around.

As long as you're discussing Selichot, maybe you'd be kind enough to answer my Selichot questions.

Shana Tova.

Keep thinking about it, TRN and all. I'll give you all the way until "early next year" before posting the answers!

Shanah tovah, Elie!

TRN - or others - any new thoughts on these or should I post the solutions?

Ah, how about a hint? I would very much like to feel accomplished in solving these math problems, but I need a push in the right direction. Can you help me figure out how to think about them?

And thank you so much for asking!

OK, here are a couple of clues:

For #1, whether the past year was a leap year or not is actually irrelevant.

For #2, in addition to the information needed to solve #1, you need to consider in which years selichos before RH start the previous Sunday, or two Sundays before.

I'll plan to post the answers on Sunday or Monday.

I have no idea how to figure out for which years Rosh Hashanah falls on which of the four days on which it can fall, and I don't know of a cycle shorter for the total number of years for those date/weekday patterns than that huge one.

I did end up reading about under what conditions we need to begin Selichos two Sundays before, so I've got that piece, I believe.

If others can help with the information needed and brainstorm with me in figuring the approach, I'd be happy to do the math. Or, working together, I'm sure we could tackle this. Soccer Dad is both skilled in mathematical thinking and knowledgeable about tefilah, no?

Or maybe I ought to just quit before I display any more of my ignorance.

TRN, it seems you might be overthinking #1 somewhat. If you know which four days RH can fall, you are partway to the answer.

Hmm.

In the first question, the exact percentage isn't required, as "about what percentage" is what is requested. I had figured this meant that the final answer could be rounded for ease. But maybe the approximation comes up earlier, when figuring out how often Rosh Hashanah falls on each of the four days? Because it certainly varies, depending on within how many and which years one counts; the cycle is not as simple as the cycle for leap years.

For the second question, which does not use "about" or another such word to modify "average," does the answer take into account the frequency, or is the average only of the four different possibilities, not weighted by how often each occurs?

For the first question: Nitzavim and Vayelech are separate when Rosh Hashana falls on Monday or Tuesday, since there are two Shabbatot between Rosh Hashana and Sukkot. Niztavim is always read on the Shabbat before Rosh Hashana (either with Vayelech or alone). If there are are two more Shabbatot, one must be for Vayelech and one for Haazinu. This occurs about 40% of the time. The other possible days for Rosh Hashana are Thursday and Shabbat, and in these situations there is only one Shabbat (Shabbat Shuva) between Rosh Hashana and Sukkot.

Okay, from what I can calculate from data I found -- do people just know this? -- it looks like Rosh Hashanah falls on Monday about 28% of the time, Tuesday about 11.5%, Thursday about 32%, and Saturday about 28.5%.

I found the number to which I had earlier alluded of years in the complete cycle, incidentally -- 689,472! I don't have data for the full cycle, and I haven't figured out how to compile it myself. I can only assume that what is true for a small fraction of that cycle is generally true throughout, which seems to be a safe assumption.

So Nitzavim and Vayelech are read individually when Rosh Hashanah falls on Monday or Tuesday, if the key is not having either Rosh Hashanah or Yom Kippur fall on Saturday.

If so, the answer to the question is Nitzavim and Vayelech are read separately compared to together about 39.5% of the time.

Now, I ended up reading that Selichos have to be said for a minimum of four days before Rosh Hashanah begins, so that we get at least ten days in, and therefore we begin two Sundays before rather than the Sunday before when Rosh Hashanah falls on Monday or Tuesday.

If I've got this right, then, we say Selichos for 13 days when Rosh Hashanah falls on Monday (a week minus a day plus a day plus the six days), 14 when on Tuesday (add another day), 10 when on Thursday (four days plus the six days), and 12 when on Saturday (add another two days).

The average of those four options is:

(13+14+10+12)/4 = 12.25

but the weighted average, using the approximate frequencies from before, is:

13*.28 + 14*.115 + 10*.32 + 12*.285 = 11.87

Wow, and while I was toiling over all that, in breezes Bruce, answering with ease! Well done, Bruce. But how did you know or figure it was about 40%?

Elie, I am eager to hear how I could have avoided overthinking this.

Thanks TRN and Bruce! I was getting ready to spell out the correct answers this morning, and you both went and saved me most of the work!

I have nothing to add to TRN's detailed answer to #2, but a couple of further comments on #1. When I mentioned "overthinking" I was referring to TRN's question on "how to figure out for which years Rosh Hashanah falls on which of the four days on which it can fall". Basically the main factor needed is that it falls on Tuesday must less frequently than the other days. Oversimplifying, this is basically because the three days that RH

cannotfall on - Sunday, Wednesday, and Friday - each are a day before theotherthree possibilities - Monday, Thursday, and Saturday respectively. Thus, under our fixed calendar, whenever RH "would" have fallen (due to the actual molad) on one of the disallowed days, it is pushed a day later, and thus it falls on Monday, Thursday, and Saturday at least twice as often as on Tuesday. I would have assumed, therefore, that it falls on Tuesday about 1/7 of the time. TRN, as your research shows, it might be even less often than that!The other key factor, to paraphrase Bruce, is that when we come to the Shabbos before RH, there are

alwaysthree sedras left to read (besides V'zos haberacha), and thus whether Niztavim/Vaylech are combined depends only on whether there is a Shabbos between YK and Sukkos. It does not depend on whether the past year was a leap year, since no matter what, Divarim is always read the Shabbos before Tisha B'av, which leads the the fact stated at the beginning of this paragraph.Thanks again guys. I'm going to post a bonus question shortly - an easier one, I think.

I must confess - I got the percentages from the OU Torah Tidbits published here in Israel. They have stats on everything!

I knew that Nitzavim and Vayelech are separate when RH falls on Monday or Tuesday, and I knew that Tuesday is the least common (it goes with Seder on Motzaei Shabbat), but I didn't know the exact percentages.

TRN - Kol hakavod on your answer to number 2.

G'mar Chatima Tova

Oh! Now I see why I was so stuck.

I was looking for a frequency based on the full cycle of years, much like the seven-out-of-nineteen figure. But I knew this cycle was very long and did not yield a simple frequency.

It didn't occur to me to start by thinking in sevenths!

I began from the point at which the standardized calendar has already been calculated and looked to it to gather the data to then determine the frequencies. You seemed to be indicating that answering the questions did not involve this complication, but I couldn't figure out how else to find the frequencies. I couldn't imagine one would simply know that information!

What you did was go back to the beginning, start with what the frequencies would naturally be with no modifications, and then apply the most frequently-invoked postponement rule. Though "Lo ADU Rosh" is not the only postponement applied and doesn't seem to be the first rule applied, and though sometimes Rosh Hashanah is shifted by two days and not only one, postponing by one day from the disallowed days gets the numbers fairly close.

I was looking for exact percentages based on the fixed calendar and did not understand that you were looking for approximate percentages based on only the most common postponement.

Of course I knew on which days Rosh Hashanah can and cannot fall and why, but I only thought of the final fixed calendar that reflects this, and not of the act of shifting from the disallowed days to make it so. I never even noticed that there aren't two disallowed days in a row.

I didn't think to look to the calculation of the calendar for the calculation of the frequencies. Of course it makes sense to start there!

It was interesting to explore all of this, Elie.

Also, the shifting of the full sevenths reminds me of the Monty Hall math problem.

Bruce, good job, and thank you for pointing out the relationship between Rosh Hashanah on Tuesday and the previous Pesach, and of course thank you for your kind words about my answer.

doesn't seem to be the first rule appliedScratch that; while true, it isn't important here. The first rule applied, the "old molad" modification, doesn't affect the frequencies, as it shifts one fourth of the "would have fallens" from each of the seven days to the next. So the frequencies are kept distributed equally until next the postponements from the disallowed days are applied.

The two other rules shift a very small perecentage from Tuesday to Thursday and an even smaller percentage from Monday to Tuesday.

So thinking in sevenths is good thinking.

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