**all**the animals. And that is why the תורה does not go into any detail concerning the division for it was not exact.

## Friday, August 2, 2024

### Splitting up the Animals

## Friday, July 26, 2024

### No Population Increase

### The Probability of the Goral

*pasuk*26:54, the Torah explains how the land was divided amongst the tribes. Rashi explains exactly how the lots were picked to determine each tribe's portion.

2^{12} |

(24 x 23 x 22 x 21...x 2 x 1) |

^{12}/24!. When all is totalled, the odds of the draw falling out exactly as planned without any Divine intervention would have been one in 151,476,000,000,000,000,000.

^{2}. The odds of the second Nasi will be 1/11

^{2}. And so on. The total odds will be:

1 |

12^{2} x 11^{2} x 10^{2} x 9^{2} x 8^{2} x 7^{2} x 6^{2} x 5^{2} x 4^{2} x 3^{2} x 2^{2} x 1^{2}^{} |

^{2}. The final odds will be one in 229,442,532,802,560,000. This is approximately 660 times more probable than the odds according to Rashi on the

*chumash*.

*goral*'s results to have been produced naturally. This is a veritable testimony to the extent of the miracle that occurred and the Divinity of the apportioning of Eretz Yisroel to the twelve tribes.

תשע"ד: A similar drawing came up this week in דף יומי on :תענית כ"ז regarding the גורל used to determine the order of the משמרות in the בית שני. According to רש"י's understanding, there was a similar miracle at play. The probability associated with that drawing is discussed here.

## Friday, July 19, 2024

### Counting the Judges

^{1}. Of course the obvious easy way out would be to say that there is an error in our version of Rashi, which would require only the replacing of the word

*shemonas*with the word

*shiv'as*to get close enough to the true number. This, in fact, seems to be the version of Rashi that the Ramban had. However, whenever this is avoidable it is best not to rely on such an answer and to justify our reading of Rashi. But is it avoidable in this instance?

Perhaps, it is possible that Rashi is not referring to the actual number given in Sanhedrin but to the calculation done there. Just as when B'nei Yisroel were 603,550 the number of dayanim was 78,600 the number of dayanim based on B'nei Yisroel's current population would be around 88,000. The next step, then, is to calculate what population would require 88,000 dayanim. Being that the dayanim included judges of groups of 1000, 100, 50 and 10 the following equation is given:

88,000 | = (x/1000) + (x/100) + (x/50) + (x/10) (where x = total population) |

= (100x + 20x + 10x + x)/1000 (multiplying by 1000/1000, i.e. 1) | |

= 131x/1000 (131 judges per 1000 citizens) | |

x | = 88,000,000/131 |

x | = 671,755(rounded down) |

We are left, then, with only three incidents to consider. The first is the episode following the death of Aharon when B'nei Yisroel began to return toward Mitzrayim and B'nei Levi ran after them (see Rashi 26:13). The Yerushalmi records that only eight (or seven, see Rashi and the Yerushalmi inside) families were wiped out from Yisroel in that incident. This would seem to represent a significant loss but perhaps not 46,000.

The second is the episode with the snakes (Bemidbar 21) where, as recorded in the Mishna (Rosh HaShanah 29a), those who were bitten and did not have appropriate

*kavana*when shown the copper snake perished. Here, too, there is no significant loss recorded but only that proper

*kavana*was required for the snake to cure you. Before the cure, however, the pasuk states (pasuk 6) that a great multitude perished from Yisroel. We are not given any further information, though, on the number of casualties

^{2}.

Finally, there is the plague of Ba'al Pe'or. That leads into a discussion as to what in fact transpired besides the loss of 24,000 in the plague. From the fact that the Torah refers to a

*magefa*, it is doubtful that this refers to those killed by the shoftim. So how many people did the shoftim kill? The Ramban (on this pasuk) quotes a Yerushalmi, the simple reading of which implies that each shofet killed two men as commanded by Moshe. This would render well over 150,000 casualties. Ramban, however, concludes that the figure given by the Yerushalmi is just referring to how many would have been killed had Moshe's command been carried out but in the end the shoftim never had a chance to do so and they didn't kill a soul. Perhaps it is possible to take this idea of the Ramban that the shoftim were interrupted before having a chance to complete their mission, but to suggest that they had already begun to carry it out when they were interrupted. With or without such a supposition, one could suggest that in some way, these three incidents combined for a grand total of 46,000 fatalities. The other 24,000 died in the

*magefa*. Now we have accounted for all 70,000 lives and all the figures work out.

Nevertheless, it is doubtful that Rashi actually wrote 88,000 and had this convoluted calculation in mind. Rather, it is more reasonable to assume that this version of Rashi is a mistake and that he originally wrote 78,000, particularly because the Ramban had such a reading of Rashi. However, I am not the first to try and justify the figure of 88,000. The Margaliyos HaYam on the gemara in Sanhedrin cites the sefer Techeles Mordechai who offers a calculation based on a population of 603,550:

603 | shoftim over a thousand |

6,035 | shoftim over a hundred |

12,071 | shoftim over fifty |

60,355 | shoftim over ten |

70 | zekeinim |

276 | (12x23 small Sanhedrin for each tribe) |

72 | (12x6 Nesi'im for each tribe) |

8,580 | Levi'im |

=88,002 |

^{1}Although it is not the subject of this piece, it is interesting to note the various discussions concerning the calculation in Sanhedrin. The Yad Ramah and one opinion in Tosafos state that the shoftim were all over 60 and were not part of the general population. Another opinion in Tosafos states that the shoftim of 50 were taken from the shoftim of ten, the shoftim of 100 were taken from the shoftim of 50, etc. This, however, is not in accordance with the Yerushalmi quoted here by the Ramban. See also Margaliyos HaYam who raises a number of interesting questions regarding the figure given there. It bothered me, though, that the calculation is based on 600,000 not 603,550 which would render a different total.

^{2}The Zohar in Parshas Balak cites an opinion that this pasuk is referring to Tzelafchad alone for he was the leader of his tribe (Source: Ta'ma D'Kra, R' Chaim Kunyevsky)

## Friday, June 21, 2024

### Piles of Quail

*parsha*, we have the episode of the quail that fell outside of the camp. The pasuk (11:32) recounts that the one who gathered the least gathered 10 mounds of quail. The GR”A has a fascinating calculation to figure out how this number was reached. It is assumed that the one who gathered the most would have been the one whose tent was at the outskirts of the camp because the quail fell outside the camp. The one who gathered the least would be the one whose tent was the furthest inside the camp. The gemara (Berachos 54b) tells us that the camp was 3 parsa by 3 parsa. Therefore, someone who lived on the very inside of the camp would have to walk three parsa in order to get a pile of quail, one and a half there and one and a half back. The gemara (Pesachim 93b) also tells us that a regular man can walk 10 parsaos in a day (not including the night). According to the pesukim, the quail was collected for a day, a night and a day, a total of one and a half days. This would give the average man enough time to walk 30 parsaos - ten the first day, ten during the night, and ten again the next day. This would allow one who lived in the centre of the camp to travel back and forth ten times. That is how the pasuk arrived at this number.

## Friday, June 7, 2024

### Tens and Ones

The first question is, why is this the case.

Furthermore, I noticed something this week that I don't recall ever noticing before: In one instance in the parasha, this style is violated. Pasuk 2:9 gives the total of the Eastern Camp, including the tribes of Yehuda, Yissachar, and Zevulun. The number is 186,400, written as follows:

```
כָּל-הַפְּקֻדִים לְמַחֲנֵה יְהוּדָה, מְאַת אֶלֶף
```**וּשְׁמֹנִים אֶלֶף**

וְשֵׁשֶׁת-אֲלָפִים וְאַרְבַּע-מֵאוֹת--לְצִבְאֹתָם; רִאשֹׁנָה, יִסָּעוּ.

"... a hundred thousand and eighty thousand and six thousand and four hundred ..."

This is a clear departure from the usual style, which would have been "ששה ושמנים אלף", "six and eighty thousand". I am not aware of any other such departure from the usual style. Any ideas why this is?

### Rounded numbers

**Parashat Bemidbar 5759/1999**

**The Census of the Israelites in the Wilderness**

**Prof. Eli Merzbach**

**Department of Mathematics and Computer Science**

**Several censuses of the Israelites are mentioned in the Torah. It is interesting that almost all the numbers listed in these censuses appear to be round numbers, i.e., without units and even mostly without tens. Of course this can be ascribed to miracle or viewed as an inexplicable random occurrence (as some have tried to do). The great commentators have rejected interpretations of this kind on the simple grounds that there are no miracles that do not have significance or purpose.**

**Another question arises when we consider all the censuses that appear in Numbers: why does the Torah have to relay the subtotals both for the various tribes and also according to their banners? Nine sums appear in Numbers, each and every one of them accurate; but what did the Torah wish to say by relaying these sums? On this question Nahmanides wrote (Num. 1:45):**

**Scriptures had to say what the total was after giving the detailed figures because Moses and Aaron were commanded to know the census of the entire people and the census of each tribe, for that is the way of kings when counting the people. But the reason underlying this commandment--why the Lord commanded it--escapes me. I do not know why they had to know the number; why were they commanded to know it?**

Here are the detailed figures and the totals of the censuses recorded in the Torah:

Numbers 1 Numbers 2 Numbers 26

(on the Plains of Moab)

Reuben 46,500 43,730

Simeon 59,300 151,450 22,000

Gad 45,650 40,500

Judah 74,600 76,500

Issachar 54,400 186,400 64,300

Zebulun 57,400 60,500

Ephraim 40,500 32,500

Manasseh 32,200 108,100 52,700

Benjamin 35,400 45,600

Dan 62,700 64,400

Asher 41,500 157,600 53,400

Naphtali 53,400 45,400

**Total 603,550 603,550 601,730**

Census of the Levites Numbers 3 Numbers 4

(1 month and up) (age 30 and up)

Gershon 7,500 2,630

Kohath 8,600 2,750

Merari 6,200 3,200

**Total Levites 22,300 8,580**

These are all the census figures given in the Torah (with the exception of the enumeration of first-borns, whose number is very small in comparison to these figures).

In the literature of the *rishonim*, or early commentators, I did not find any direct attempt to deal with the two questions which I raised (why the figures are rounded, and what use there is in the totals or sums). *Aharonim*, or later commentators, however, struggled with these questions. In *Meshekh Hokhmah* Rabbi Meir Simha ha-Cohen of Dvinsk addressed the first issue as follows (Num. 3:16):

- Perhaps Scripture says regarding the number of Israelites, "You ... shall record them by their groups, from the age of twenty years up, all those in Israel who are able to bear arms" (1:3), because they were counted not in units, but only by tens. Therefore, none of the numbers has units, because each head of Israelites gave his number of men, and they were heads of tens. Every small number was rounded, and fewer than ten did not have a head over them alone. Therefore it says "by their groups," for no camp has fewer than ten, as explained in the Jerusalem Talmud, at the end of the first chapter of
*Eruvin*.Regarding the number of Levites, from age one month up--Moses entered the Tent and a divine voice called out and said, "Thus and so many babes," also not counting by individuals, for the count was "as he was bidden" regarding the number of Israelites [no less than ten]. But this was not the case with the list of first-borns, where each individual was reckoned.

*Shirat David*(Num. 26):

- Above, in Parshat Bemidbar (1:25) we cited
*Imrei Noam*to the effect that the reason "units and tens" were not mentioned in the counting, and each tribe was reckoned in hundreds except for the tribe of Gad (45,650), is that Scripture is not strict about the few. There we explained that the reason must be that the Torah completed the counting to the nearest hundred, and that one should not interpret that it rounded to the nearest ten, for if so how does one explain the improbable fact that no tribe had a number ending in tens save for the tribe of Gad? As for the tribe of Gad not being rounded to a hundred, that is because their number came exactly to fifty, and it was not possible to round it to a hundred since it falls just in between.But our explanation is problematic, since here in chapter 26, all the numbers are rounded out to the nearest hundred save for Reuben (43,730) ending with thirty. Clearly in the reckoning here the Torah did not complete to the nearest hundred. If so, it is wondrous how they all ended with hundreds save for one; so this must be studied further.

At least a partial resolution of the problem is provided in *Emet le-Ya'akov*, written by Rabbi Jacob Kaminetsky:

- In my humble opinion, the counting was done by the chieftains of fifties, since we see in Parshat Jethro that the leaders were divided into heads of tens, heads of fifties, heads of hundreds, and heads of thousands, and apparently the army was divided into heads of fifties. Likewise, we see in the beginning of II Kings (1:9-10) that there were captains of fifty with their fifty men. It was by these captains that the Israelites were counted, and hence there were either complete hundreds or fifties. Except that this theory encounters a problem in Parshat Pinehas (26:7), where the tribe of Reuben totals forty-three thousand, seven hundred and thirty. Possibly the Torah subtracted those of Korah's followers who were swallowed in the earth from the rounded-out fifty, and since their number was twenty that left exactly thirty, and this accounts for the exact number in 26:7, but nevertheless the matter needs more thought.

In my opinion both questions can be answered by relying on the following general rules that pertain to fairly large numbers (certainly to numbers greater than 5,000).

1) When the number obtained was in tens (with no units), then it was registered as is and the Torah did not round it.

2) When the number obtained was not in complete tens, it was rounded to the nearest hundred.

There is a simple logic to these rules: if you round a number that ends in units, then it is rounded to hundreds (the error being less than a hundredth), but a number that ends in tens is left as is. It should be noted that the simple notion which we understand of rounding numbers to the nearest hundred was totally foreign to science until the end of the Middle Ages. Otto Neugebauer, in "The Astronomy of Maimonides and its source," *HUCA* 22 [1949], p. 340, notes that also ancient astronomers who were expert in complicated computations and who regularly used rounding did not generally round to the nearest whole number. Rounding was generally done downwards, unless the number was very close to the larger number (e.g., greater than 0.75). Neugebauer stresses that Maimonides in his astronomic computations to determine when the new moon occurs rounded to the closest integer and that this was a major innovation in comparison with his predecessors such as Ptolemy or even Al-Battani.

Now let us return to the census in the Torah. As we have said, the numbers were rounded according to the two rules I mentioned above. If we look at the figures in the Torah, this is patently clear. In each of the two censuses of the Israelites in the wilderness, 11 out of 12 figures are multiples of hundreds, whereas one (in the first census the tribe of Gad, and in the second census the tribe of Reuben), is a multiple of ten. The probability of any number ending in zero but not being a multiple of one hundred is 9/100, therefore if one takes any 12 numbers, the expectancy of such a number appearing is equal to 12 x 9/100 = 1.08. In other words, on the average, out of 12 nu, one will be a multiple of ten (but not of a hundred). Moreover, if we compute the different probabilities (according to binomial distribution), it turns out that the greatest probability is obtained when exactly one out of twelve numbers has this form. The probability of this equals 12 x (1-9/100)11 x 9/100, and all the other probabilities are smaller.

Examining both censuses together also yields the same results: out of 24 figures, the average number of occurrences of the specifically desired form is close to 2, and the maximal probability is obtained when k=2, which is indeed what happened.

As for the censuses of the Levites, similar results can be obtained, but with a small number of figures (there being only three families) no statistical analysis can be made.

The rules that we used enable us to answer the question about the sums. Now it is clear why the Torah had to write down the total sums of Israelites in both censuses. Since all the numbers were rounded, one could have had a situation where the grand total obtained would be far off from the actual number in the census. In theory, for the census of the Israelites the deviation could be as great as 588 people. For example, if the number of people counted in each tribe ended in 49, then the numbers would be rounded down to the nearest hundred, so that after totalling all twelve tribes one would have a figure smaller by 588 (actually by 600, after rounding) than the actual census count. Of course this is a rather extreme example, and actually there is a mathematical theorem stating that as the number of figures being summed increases, the deviations resulting from rounding are more likely to offset one another. Actually that is precisely what happened with the census of the Israelites. All the deviations, both upwards and downwards, counterbalanced so that the sum matched the total census taken (of course, to the nearest 50), and therefore it was very important that all these figures and sums be reported in the Torah.

Prepared for Internet Publication by the Center for IT & IS Staff at Bar-Ilan University.

### Discrepency in לוי's Population

### Explaining the Uncounted לויים

### What are the odds?

- Let's assume that the child themselves is a ישראל, otherwise it's a non-starter. So we need to know the odds of their spouse not being a Levite (80% based on my snooping of our shul's membership database.)
- The first fetus has to be male (let's just say 50%)
- The baby must be delivered and not miscarried (let's use 90%)
- The baby must be born without a Cesarean (again, 90%)

## Friday, May 17, 2024

### Omer Counting in Different Bases

**not**do that. However, it was a very interesting concept I had never thought of before. So, I added a widget on the blog's sidebar which will display the day of the Omer in various relevant bases.

## Thursday, March 14, 2024

### Happy π Day

Here are 10 ways to celebrate Pi Day, including this young chap who memorized 2,552 digits (eat your heart out, Brodsky.)

## Friday, February 16, 2024

### עמודי החצר

*amos*between each beam and 20 beams, that would provide only 19 spaces of five

*amos.*That would yield only 95 of the 100

*amos*that the pasuk tells us make up the length of the courtyard. Of course, the first notion is that the space does not include the width of the beams. Therefore, there may have been 95

*amos*of space and five

*amos*of beams to complete the 100

*amos*. This is in fact the suggestion of the Riva, in the name of his rebbe and is also the opinion of the Abarbanel. The 20 beams on the north and south sides added up to five

*amos*on either side. This would make each beam one quarter

*amah*(1½

*tefachim*). This interpretation would avoid all our problems from the outset. However, R' Eliyahu Mizrachi takes issue with this interpretation on two accounts. Firstly, he sees no reason why there should be such a large difference between the thickness of the beams of the courtyard and that of the planks of the Mishkan itself (nine

*tefachim*). His second objection is that within the beams themselves you would have some of different thickness than others. On the east and west sides, there are only 10 beams needed to make up five

*amos*. (The nine spaces between the ten beams make up 45 of the 50

*amos*width of the courtyard.) Therefore, each beam would be three

*tefachim*, twice the width of those on the north and south sides. The lack of symmetry involved in this understanding of Rashi causes the Mizrachi to disregard it and give his own interpretation.

*amos*referred to by Rashi are not five

*amos*of space but rather five

*amos*from the beginning of one beam to the beginning of the next.. This view is generally accepted amongst all those who deal with this problem with the obvious exception of the aforementioned Riva and Abarbanel. In pasuk 18, the Mizrachi infers from Rashi that the beams were one

*amah*thick. Therefore, the actual space between each beam would be four

*amos*and the thickness of the beam would complete the five

*amos*. However, we have now only accounted for 95

*amos*. Therefore, the Mizrachi suggests that the north and south sides actually had 21 beams and the east and west had 11 but that the seemingly extra beam on each side belonged to the set of of beams of the side perpendicular to it. For instance, 21 beams were placed on the southern side of the courtyard. The beam in the southwest corner, though, was officially part of the western side. So, too, the beam in the northwest corner was not counted as part of the western beams but as part of the northern beams and so on. See illustration. With this arrangement another space of five

*amos*is added to complete the 100

*amos*referred to in the pasuk.

*amah*measurement of the courtyard was in fact a measurement from within the beams and the one

*amah*taken up by the beams is not included. This reasoning was given in order to justify Rashi's calculation of 20

*amos*distance between the Mishkan and the curtains of the courtyard on the north, south and west sides. The Gur Aryeh objects to this with the claim that the pesukim (9,11,12,13) clearly state that the curtains were exactly 100

*amos*long on the north and south sides and 50

*amos*long on the east and west sides. But according to the Mizrachi's interpretation, the outer perimeter of the courtyard would be 102

*amos*by 52

*amos*. He offers a defence for the Mizrachi that perhaps the only purpose of the curtains was to cover up the open spaces and they did not need to cover the corners (as illustrated on page 3). However, in his own opinion, the Gur Aryeh suggests that the 100

*amah*measurement is in fact referring to the outer perimeter of the courtyard. He then was required to justify Rashi's measurement in pasuk 18 in a different manner.

*amos*between each beam more than just once, he infers that Rashi meant for this to be consistent throughout the entire perimeter of the courtyard. According to the Mizrachi the length of the north side, for instance, was really 102

*amos*and according to the Gur Aryeh it was 100. However, if you add up 21 beams each of one

*amah*thickness and 20 spaces of four

*amos*each, we are given 101

*amos*. So, too, on the east and west sides we would end up with 51

*amos*instead of 50 or 52. He concludes that the only way for the Mizrachi's figures to work out would be to say that one space on each of the four sides was actually one

*amah*bigger. For the Gur Aryeh's figure to work one space would have to be one

*amah*smaller. The Levush does not accept that such a lack of symmetry was present in the building of the Mishkan and offers a rather unique arrangement of the beams. Each of the beams were circular on the bottom for one

*amah*and were inserted into circular holes in the copper sockets that held the beams in place. The beam itself was a semi-cylinder whose diameter was one

*amah*. On each of the corners was placed a quarter-cylinder beam so that the curtain could wrap around it. See illustration. The thickness of this beam was only one half

*amah*on either side. This removes one half

*amah*one either end of each side of the courtyard. With this arrangement, the spaces between all of the beams were all four

*amos*wide without any exception and the perimeter of the courtyard was exactly 100

*amos*by 50

*amos*as stated in the pesukim. Amongst all the interpretations mentioned thus far, this is by far the most symmetric and arithmetically accurate.

*amos*long. The north and south sides had twenty such bars and the east and west sides had ten. These wooden bars would allowed the curtains to change direction at the corners without the need to wrap it around a beam. See illustration. Once again the figure of five

*amos*refers to the distance from the beginning of one beam to the beginning of the next. With this arrangement the thickness of the beams becomes irrelevant. All of the figures mentioned in the pesukim work out perfectly as well. One advantage of this arrangement over that of the Levush's is that all of the beams are the exact same shape.(The illustration assumes the beams to be one

*amah*thick.)

*amos*between each beam, he had some logical calculation in mind. The only question that remains is "Which?".

**On a Related Topic**

The Mishkan was covered by three layers of material(*). The first covering described by the Torah (26:1-6) was made of twisted linen, turquoise, purple and scarlet wool. The covering was made up of 10 panels of 4x28 amos2. This yields a total area of 40x28 amos2. The Mishkan was 30x10 amos2. The beams that made up the walls of the Mishkan were 1 amah thick. Thus, the Mishkan required 32x12 amos2 of roofing.

The beams were 10 amos tall. The covering was 28 amos wide and 12 amos covered the roof of the Mishkan. That leaves 16 amos for the two sides which is 8 amos on each side. So the wool/linen would reach two amos from the ground. There is a dispute as to whether or not the front beams were covered. We will go with the opinion of the gemara (Shabbos 98b) that they were uncovered as Rashi (26:5) notes that the pesukim seem to indicate as such. Therefore, 31 amos of the covering's width provided roofing, leaving 9 amos to hang from the back. The second covering was a covering of goat hair. This covering was wider and longer than the wool/linen layer and covered it fully on all sides.

Rashi (26:13) notes that the Torah teaches us a lesson that one should show compassion for valuable objects. The twisted linen and assorted wools were very precious and thus, as Rabbeinu Bachya explains, it was made not to drag on the ground so that it would not be soiled by dirt and rain and was protected fully by the goat hair. This lesson is easily understood considering the measurements mentioned thus far. However, there is one simple question to be asked. What about the corners? As the accompanying diagram shows, if a piece of material hangs only 8 amos off one side and 9 amos off the other, simple Pythagorean geometry dictates that the corners will hang down more than 12 amos! (This effect is well demonstrated by the corners of a rectangular tablecloth hanging from the table.)This is hardly an efficient way to care for valuables.

This problem seems far too obvious to have been overlooked by Chazal in teaching us this lesson. However, finding the answer was not easy. But finally, an answer was found in R' Chaim Kunyevsky's elucidation of Braisa diMleches haMishkan. There he asks exactly this question. He answers that the corners of the coverings were folded against the back of the Mishkan as illustrated. The Ritv"a (Shabbos 98b) apparently provides the same answer in the name of Braisa diMleches haMishkan but our versions show no evidence of any such discussion. One of the books on the Mishkan actually show such an arrangement but there is no discussion as to any source or reason for it.

*This and a number of other facts discussed on this page are actually subject to a large-scale dispute between R' Yehudah and R' Nechemiah. For our purposes, all figures are according to R' Yehudah.

## Friday, January 26, 2024

### חמושים

**could have been**would be 30 million, still a 1:5 ration. I don't see how the fraction can be interpreted the way R' Schwab did. Nevertheless, it is the only answer I've seen to these difficulties with the Midrash.