.י"ג Weight a Minute

אם על המלך טוב יכתב לאבדם ועשרת אלפים ככר כסף וגו' אמר ריש לקיש גלוי וידוע לפני מי שאמר והיה העולם שעתיד המן לשקול שקלים על ישראל לפיכך הקדים שקליהן לשקליו והיינו דתנן באחד באדר משמיעין על השקלים ועל הכלאים

תוס' ט"ז. ד"ה דחי עשרה אלפי ככרי כספא. שמעתי שעשרה אלפי ככר כסף עולין חצי שקל לכל אחד מישראל שהיו שש מאות אלף כשיצאו ממצרים ואמר שיתן לאחשורוש כל פדיונם (ה) ודוק ותשכח:

A fascinating piece attempting to unlock the mystery of this תוספות.
(Get out your calculators.)

.ב No More, No Less

מתני' מגילה נקראת בי"א בי"ב בי"ג בי"ד בט"ו לא פחות ולא יותר

How about a nice piece of דרוש to start off the מסכתא. From the גר"א in שדה אליהו:

If you add up the גמטריא of the days on which the מגילה can be read, it equals 65. (Sorry if you were already in the middle of using the Gauss method.) This is the same as the גמטריא of השם’s name, א-ד-נ-י. This name is associated with השם’s more hidden Providence which, as we know, was behind the story of פורים.

 
But here’s where it gets cute. The name י-ה-ו-ה is associated with השם’s more blatant and obvious השגחה, which we did not merit to see in the story of פורים. The גמטריא of that name is 26. The משנה states that the מגילה cannot be read before the 11^th or after the 15^th. Therefore, the days on which it cannot be read are 10 and 16. Add them up and what do you get? 26!!

:כ"ז Similar Raffles

וחלקום והעמידום על עשרים וארבעה בללום ונתנום בקלפי בא ידעיה ונטל חלקו וחלק חבריו שש בא [חרים] ונטל חלקו וחלק חבריו שש וכן פשחור וכן אימר

How fitting that this גמרא falls out in פרשת פינחס. In an installment of על פי חשבון, I attempted to calculate the probability of the Divinely "rigged" גורל falling out exactly like it did. So, then, what would be the odds of this one coming out like it did without the obvious Divine intervention?

I have to admit, I'm not sure I fully understand the exact process and exactly who picked what. I also have not investigated the מפרשים sufficiently. So I will keep it as simple as possible. Therefore, we will assume that the one picked for each משמר did not have to pick the name of the משמר first. In other words, ידעיה picked first. He came up with 6 names, all of which belonged to the ידעיה family but one was actually named ידעיה. Let us assume that he picked all 6 at the same time*. Using simple nCr notation, there would be ₂₄C₆ different possible combinations for ידעיה. For the uninitiated, that is simply a fancy way of expressing the following: 24! / (24-6)! x 6! Simply put, the different possible combinations ar

24 x 23 x 22 x 21 x 20 x 19

6 x 5 x 4 x 3 x 2 x 1

The reason we have to divide by 6 x 5 x 4...  is because order doesn't matter, so there are that many different combination which are essentially one in the same.

I probably have not sufficiently explained the above concept for those who are not familiar but if we move on, I believe the probability can be expressed thusly:

1

24C6 x 18C6 x 12C6

The probability of the final group is essentially 1 since there are only those 6 tickets left so we really only need examine the probability of the first three groups. That computes to

1

134596 x 18564 x 924

Which is 1 in 2,308,743,493,056. Although both this and the גורל of פינחס both involved 24 tickets, since this one was drawn in groups of 6, the probability is in fact nearly 100,000 greater than that of פינחס. Nevertheless, as was discussed in my parallel calculation of פינחס, the odds would still be close those of getting fatally hit by lightning twice in one year! Again, this is a veritable testimony to the extent of the miracle that occurred and the Divinity of the raffle.

*Well I just looked a little deeper into this.:רש"י ערכין י"ב explains that the raffle happened exactly the way I just assumed that it didn't. So first the head of the משמר picked his own name and then the other 5. This complicated things slightly. The probability would therefore be:

1

24 x 23C5 x 18 x 17C5 x 12 x 11C5

Which - let's just skip to the end - is 1 in 498,688,594,500,096, or over 200 times less probable than originally thought.

You want to simplify all this? According to רש"ש, there was nothing particularly miraculous going on here at all. There were in fact 4 separate raffles of 6 whereby each of the 4 משמרות would draw to determine the order of their respective subdivisions.

.ח Easy as Pi

אמר ליה רב אסי לרב אשי לעולם גברא באמתא יתיב ורבי יוחנן מקום גברי לא קחשיב כמה הוו להו תמני סרי בשיבסר נכי חומשא סגיא היינו דלא דק ולחומרא לא דק

A "mathy" guy like myself can't help but be curious to examine the precise math behind this גמרא. The results are quite fascinating.

Let's start with what the precise הלכה would be:

If we do, in fact, require a circle that encompasses a 4x4 square, the radius of that circle would be √(32) by the Pythagorean Theorem. That would make the circumference of that circle √(32)π which is approximately 17.7715 אמות circumference. 

Now, let's examine ר' יוחנן's statement according to רב אסי. The גמרא assumes based on its loose interpretation of π that if we made a circle of 24 men, the circle inside of that ring would be 18 אמות circumference. We could end right here and declare that the margin of לא דק is even more narrow than the גמרא suggested. But let's try to get a little more precise. A circle of 24 אמות circumference would have a diameter of 24/π. The men take up exactly one אמה so the inner circle would then have a radius of 24/π - 2. The circumference of that inner circle would then be (24/π -2) x π which comes out to 17.7168 !!! So ר' יוחנן's statement is off by only 0.0547 אמות !!! The only issue, of course is that it is לא דק לקולא but at that minuscule amount, that should be allowable. 

:ז Pi in the ... תורה

רש"י ד"ה באמתא יתיב. כל אחד מקומו אמה נמצא הקיפה כ"ד אמות וקיימא לן (עירובין דף יג:) כל עגול שיש בהקיפו ג' טפחים יש ברחבו טפח דכתיב (דה"ב ד) בים שעשה שלמה עשר באמה משפתו אל שפתו עגול סביב וקו שלשים באמה יסוב אותו סביב לעשר אמות רוחב שלשים אמה היקף אלמא הכא נמי בתריסר אמות היקף סגי דנהוי פותיא ארבעה:

Since רש"י references the ים של שלמה, it would seem appropriate to reference the גר"א's famous פשט on that.
Read more here.

:ח A Pythagorean Question

אמר רב כהנא הואיל ושמעתתא דכהני היא אימא בה מילתא הא דאמרת מניח הקורה באלכסון לא אמרן אלא שאין באלכסונו יותר מעשר אבל יש באלכסונו יותר מעשר דברי הכל אינו מניח אלא כנגד הקצר

רש"י ד"ה שאין באלכסונו יותר מעשר. שהמבוי קצר לרחבו הרבה ואין ברוחבו ובארבע אמות הנמשכין להצטרף יותר מעשר

This line in the גמרא seemed fairly simple to me. However, I found the way רש"י explained it to be somewhat troublesome. If I'm understanding correctly, he is suggesting that the מבוי would need to be extremely narrow in order for the אלכסון to be less than 10 אמות. But the simple Pythagorean calculation suggests otherwise. By the Pythagorean Theorem, as illustrated below, a 10 אמה diagonal is achieved only with an approximately 9.2 אמה width, given the extra 4 אמות of wall. A מבוי anyways has to be less than 10 אמות wide. So what exactly does Rashi mean?