## Wednesday, December 20, 2023

### Can you Count to 70?

## Friday, December 1, 2023

### Goats and Amicable Numbers

In this week’s parasha, we find Yaakov preparing for his encounter with his twin brother Esav in several ways. Among other preparations, Yaakov sends him gifts consisting of various different kinds of animals. The Torah tells us (Bereishit 32:14–16) how many of each kind of animal Yaakov sent: 200 female goats and 20 male goats; 200 female sheep (ewes) and 20 male sheep (rams); 30 nursing camels with their young; 40 female cows and 10 bulls; 20 female donkeys and 10 male donkeys. What is the significance of these numbers?

In his ספר בעלי ברית אברם, R’ Avraham Azulai provides an explanation for the number of goats, which he attributes to R’ Nachshon Gaon of the 9th century. The total number of goats is 200 + 20 = 220. What significant property does the number 220 have?

Consider the factors of 220, that is, numbers that multiply together to give the product 220. We can factor the number 220 in the following ways:

220 = 1 × 220

220 = 2 × 110

220 = 4 × 55

220 = 5 × 44

220 = 10 × 22

220 = 11 × 20

Now consider only the “proper factors” of 220 – that is, all the factors in the above list, excluding the number 220 itself – and add them up:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

So the proper factors of 220 add up to 284.

We now repeat the process, considering the factors of 284. We can factor the number 284 in the following ways:

284 = 1 × 284

284 = 2 × 142

284 = 4 × 71

Again, we consider only the proper factors of 284 – all the factors in the above list, excluding the number 284 itself – and add them up:

1 + 2 + 4 + 71 + 142 = 220

So the proper factors of 284 add up to 220. Does this number look familiar?

As we have just shown, the numbers 220 and 284 have the property that the proper factors of each number add up to the other number. A pair of numbers with this property is known as a pair of

**amicable numbers**, or according to R’ Nachshon Gaon, מנין נאהב. Apparently it was known to the ancients that in order to gain the love of kings and princes, a person would give one of a pair of amicable numbers as a present, keeping the other number for himself. This is so that the factors of the number given add up to the number kept, and the factors of the number kept add up to the number given. So this is what Yaakov did. He sent Esav 220 goats, and kept 284 for himself.

Wait a minute: The Torah tells us that Yaakov gave Esav 220 goats, but where do we see in the Torah that he kept 284 for himself? Several pesukim later, as Yaakov gives instructions to the servants carrying the gifts, the Torah records (32:21), “כי־אמר אכפרה פניו במנחה ההולכת לפני” – “for he said: I will win him over with the gifts that are being sent ahead.” R’ Nachshon Gaon explains that this sentence contains a hint to the number 284, in the following way. The word אכפרה can be divided in two parts: אכ פרה. When the Torah uses the word אך, it is generally interpreted by the Rabbis to indicate exclusion or reduction. Calculating the numerical value of the second part of the word, פרה, we get: 80 (פ) + 200 (ר) + 5 (ה) = 285. Applying a reduction (indicated by אך) to the value 285 (given by פרה), we obtain a value of 284. This represents the number of goats that Yaakov kept for himself, according to R’ Nachshon Gaon.

Special thanks to Daniel Levenstein for bringing this insight to my attention.

Addendum: Yaakov sent a number of different types of animals. Why were only goats sent in amicable numbers? See an interesting thought from רבנו בחיי which may shed some light on this question.

References:

Leonard Eugene Dickson, History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington: Washington, 1919, p. 39, available at:

http://www.archive.org/stream/historyoftheoryo01dick#page/38/

ר' אברהם ב"ר מרדכי אזולאי, ספר בעלי ברית אברם, published 1873 but existed in manuscript for 300 years previously; pp. 48–49, available beginning at:

http://www.hebrewbooks.org/pdfpager.aspx?req=3997&pgnum=47

הנאהבים והנעימים - על רעות אצל מספרים, in Michlalah Jerusalem College's mathematical journal אלף אפס (ℵ₀):

http://alefefes.macam.ac.il/article/article.asp?n=15

(may not work in all browsers)

(Thanks to Yaaqov Loewinger for this link via Hebrew Wikipedia)

## Friday, September 1, 2023

### Balancing the Shevatim at Har Grizim and Har Eival

*pasuk*tells us (27:12-13) that the tribes of Shimon, Levi, Yehuda, Yissachar, Yosef and Binyamin stood on Har Grizim for the delivering of the

*beracha*. Reuven, Gad, Asher, Zevulun, Dan and Naftali stood on Har Eival for the delivering of the

*kelala*. The

*gemara*(Sotah 37a) presents a quandary based on a

*pasuk*in Yehoshua that seems to show that the

*kohanim*were in the middle of the two mountains. So how could they be said to have been on Har Grizim? The

*gemara*gives three different answers as to how the

*kohanim*were split up, some below, and some on the mountain (according to two of the answers.) The answer that seems to be most dealt with amongst the

*meforshim*is that those who were 'fit for work' were below with the

*aron*, and those who were not were above. Rashi learns this to refer to those above thirty while the Maharsh"a learns that it is referring to

*b'nei*Kehas who were in charge of the

*aron*.

*gemara*that we have discussed, if you examine every single possible formation of the tribes, the actual formation of the tribes is the absolute most even division of the tribes possible. That is, the difference in population between the two mountains is at a minimum with this formation. [I personally wrote a computer program to test it out and it worked. In the article, they include a list of all possible combinations and their respective differences.] What is even more fascinating, is that this works out for both Rashi and the Maharsh"a. And what may be the most fascinating of all is that according to the Maharsh"a, the population on Har Grizim would have been 307,929 and that of Har Eival 307,930. No, that's not a typo. That is a difference of 1! According to both, this is by far the most even division of the tribes.

IMPORTANT UPDATE: Tradition magazine has been gracious enough to make their archives fully available to the public! So following the links above will now allow you to read the articles in their entirety.

## Friday, July 28, 2023

### Moshe's pleas

### Gematrias off by 1

*gematria*, finding a significance in the numerical value of a word or group of words. The Steipler Rav devoted the back of his

*sefer*, ברכת פרץ, to

*gematrios*on the

*parsha*that can blow the mind. These aren't simply one word equaling another. Time after time he will find a phrase in the Torah having equal numerical value to the phrase that Rashi uses to explain it. One of the well known rules of

*gematrios*is that it is allowed to be off by one. What the deeper reason is for this, I do not know. However, in the הקדמה to the לקט יושר, a fascinating proof to this concept is brought from the

*gemara*, in the name of the תרומת הדשן. It is not only pertinent to this week's

*parsha*, it is also connected to Tisha B'Av which we recently commemorated, hopefully for the last time.

*parsha*, begins (4:25) "When you have children and grandchildren, and you dwell long in the land..." the pasuk goes on to explain that Bnei Yisrael will commit grave sins. And HaShem vows that Bnei Yisrael will subsequently be wiped out. The

*gemara*(Gittin 88a and Sanhedrin 38a) learns from a

*pasuk*in Daniel 9:14 "HaShem hastened the calamity and brought it upon us, for HaShem our God is just in all His deeds..." Is it because HaShem is just in all His deeds that he brought calamity upon us? The

*gemara*explains that if Bnei Yisrael had dwelled in Eretz Yisrael for the numerical value of the word "

*venoshantem*" (and you will dwell long), 852, then HaShem would have had to fulfill "

*avod toveidun*," you shall surely perish. However, from the time that Bnei Yisrael entered Eretz Yisrael until they were exiled was only 850 years. HaShem graciously exiled us early so that we would not be doomed to being wiped out. The question is, if HaShem was being so gracious, why didn't He at least wait one more year? It must be, therefore, that 851 would have been considered equivalent to 852 and HaShem therefore had to exile us two years before. From here we see that a

*gematria*may be off by one.

## Thursday, July 13, 2023

### Splitting up the Animals

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

## Thursday, July 6, 2023

### 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.

## Wednesday, June 28, 2023

### 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 9, 2023

### 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.

## Monday, May 15, 2023

### 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 12, 2023

### Ironic Observation

## Tuesday, April 11, 2023

### 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.