The Constant Rate of Recession

No, this has nothing to do with the American economy. There is another difficulty with the calculation that Rashi uses to conclude that the teiva was submerged 11 amos. How could Rashi base his calculation on the depth of the water decreasing at a constant rate. One can generally assume that when water decreases, it does so at a constant rate of volume. However, mathematically, if the volume of a sphere decreases at a constant rate, the rate of change of the depth will increase as the waters become shallower. The shallower the water gets, the faster it will decrease depth-wise. How then could Rashi assume that the depth decreased at a constant rate? This is the question posed by מהרי"ל דיסקין. He gives his own answers to this question. One, for instance, is that the waters receded, the ground became more saturated which slowed down the overall receding process and hence balanced out the constant rate of change of depth. But a rebbe of mine from Yeshivas Ohr Yerushalayim posed this question of none other than Nobel Prize winner Yisrael Aumann. He answered simply that mathematically, none of this is needed. True, the rate of change of depth is not directly proportional to the rate of change of volume. However, considering the size of the globe, the difference between the two within the scope with which we are dealing is negligible and would not affect Rashi's calculation. Is this true? The short answer is "Yes". The longer answer requires a little Calculus.

The radius of Earth is 6372500 m. To make things simple we will convert this to amos. Instead of using three separate measures of the amah, we will keep things neat and use an average figure of 53 cm. (6372500 ÷ 0.53 = 12023585) That translates to 12023585 amos. To make things simpler, we will round it off to 12000000 amos. This will have little effect on the final outcome. This figure will be called r_w.

The standard equation for volume:

V_w= 4/3πr_w³

Through implicit differentiation:

ΔV= 4πr_w² Δr,

where ΔV is the rate of change of volume and Δr is the rate of change of radius. We have already set r_w to be 12000000 and Δr is ¼ (amos/day according to Rashi). Therefore,

ΔV= 4π (12000000)² ¼

ΔV= 4.524 x 10¹⁴ (constant)

The goal of these calculations is to see whether or not Δr changes significantly over the course of the decreasing of the water. To see how much Δr changes, we must switch around the equation to define Δr and instead of using the figure of 12000000 for the radius, we will use the new radius when the top of the mountains became visible, 11999985.

As stated before, ΔV= 4πr² Δr

Therefore, Δr₂ = ΔV/ 4πr_new²

Δr₂ = 4.524 x 10¹⁴/ 4π(11999985)²

Δr₂ = 0.2500006250012

This means that if the waters were receding at a rate of change of depth of 0.25 amos per day when they began receding, then 60 days later they were receding only 0.0000006250012 amos/day faster, a rather negligible amount indeed.

QED

The Weight of the Teiva

If I were such a prolific author that I would have a magnum opus, I suppose this would be it. To this day, there are still people who identify me as "that guy who wrote the thing on the teiva."

It is told that one year, on a 12th Grade chumash test, Rav Moshe Heinemann שליט"א asked his students how to calculate the weight of the Noah's Ark. He did not ask for an answer, he simply asked how one would go about figuring it out. These are the calculations. And the answers:

Later on in the Parsha, (8:4), Rashi calculates based on the rate at which the waters of the flood receded, that the ark was submerged 11 amos in the water. A variety of commentaries deal with the calculation cited by Rashi and its validity, most notably the Ramban. The Sifsei Chachamim quotes the Nali"t as saying that the figure of 11 amos is only a minimum but it could have been more. There are a number of problems raised with different aspects of the calculation, some of which will be dealt with later on. Nevertheless, if the words of Rashi are taken at face value, they hold within them the key to unlocking this mystery. With the application of a single principle, the weight of the ark can be calculated. The law required for this calculation is Archimedes' Principle which states that the weight of a body floating in water is equal to the weight of the water it displaces. The ark's virtually cubic structure (according to Rashi) makes the measurement of water displacement easy to achieve. The ark was 300x50x30 amos³ in volume (Breishis 6:15). Therefore, the water displaced by the ark was 300x50x11 = 165,000 amos³. 

The next step, of course, is to convert the figure of cubic amos into conventional measures. Unfotunately, we are unsure as to the exact measure of the amah. There are three primary opinions amongst the contemporary poskim as to the actual length of the amah: Chazon Ish, R' Moshe Feinstein and GRA"CH Noeh. Because of this disagreement, they will differ on the measure of the ark's water displacement and therefore, the final figure for the weight of the ark will be different according to each. The following is a chart calculating the water displacement in cm³ based upon each of the opinions.

Metric to Imperial conversion table below

	Chazon Ish	R' Moshe Feinstein	GRA"CH Noeh
Length of amah	57.66 cm.	53.98 cm.	48 cm.
Volume of cubic amah (length/100)3	0.192 m3	0.157 m3	0.111 m3
Calculation	= 165000 x 0.192 ≈ 31630	= 165000 x 0.157 ≈ 25950	= 165000 x 0.111 ≈ 18250
Water Displacement	31630 m3	25950 m3	18250 m3

Now that we have determined the amount of water displaced by the ark, all we have to do is calculate how much that water weighed. Then by Archimedes' Principle we can assume that the ark weighed the same amount. This, however, is not necessarily so simple. The density of sea water is slightly more than that of regular water at approximately 1025 kg/m³. This figure usually remains about the same, without significant deviation, regardless of the exact temperature. The only drastic changes are observed when the water reaches extreme conditions such as freezing or boiling.

The first difficulty encountered is that during the initial 40 days of the flood, the waters were boiling hot (Rosh HaShanah 12a). This would change the density of the water substantially and consequently interfere with the calculation. However, it is important to note that Rashi's calculation is based on the rate at which the water receded after the 150 days which followed the 40 days of destruction. By that time, the waters had calmed down and most probably dropped to a more moderate temperature. Therefore, it can be assumed that the temperature of the water is a negligible factor in the calculation of the water density. However, what does seem problematic is that Rashi brings in the figure of 11 amos in 7:17 when the waters were at their highest intensity. It is almost certain that the density of the water at this point was much less than it was 190 days later. If the ark was calculated to have been submerged 11 amos by a calculation based on cooler waters, that figure should presumably be greater at the time of the actual flood.

The next issue of question in this calculation is the fact that the water was not necessarily pure sea water. It is suggested in Rashi (6:14) that the water contained sulfur. The presence of this sulfur and whatever other solvents in solution with the water could change the density of the water and affect the accuracy of the calculation greatly. This is only a problem, of course, if the words of Rashi are taken literally. The Sifsei Chachamim seem to suggest that what Rashi means is that the sulfur caused the heating of the water. Even if the interpretation is as originally perceived, it is possible that the ratio of solute to solvent was such that it would not have affected the density anyway. Therefore, for the purposes of this calculation I have chosen to ignore whatever effects the sulfur could have had on the water density and thus we are left with approximate figure of 1025 kg/m³. Based on this figure, these are the final calculations of the weight of the ark according to the three aforementioned opinions:

Chazon Ish	R' Moshe Feinstein	GRA"CH Noeh
31630 m3	25950 m3	18450 m3
x 1025 kg/m3
32420750 kg	26598750 kg	18706250 kg

In conclusion, considering the relevant opinions, it would appear that the ark weighed somewhere between 18 and 33 thousand metric tons. In comparison with other famous ships, the Queen Mary weighed 73,850 tons. It was 309 m long, about twice as long as the ark. The Titanic weighed approximately 42,000 tons. Of course, this refers to the weight of those vessels without anyone inside whereas the above calculation for the teiva included the inhabitants.

Table of Metric Conversions
57.66 cm	=	22.7 in.
53.98 cm	=	21.25 in.
48 cm.	=	18.9 in.
31630 m3	=	1117003 ft3
25950 m3	=	916416 ft3
18450 m3	=	651556 ft3
25o C	=	77o F
1025 kg/m3	=	2260 lb/61024 in3(35.3 ft3)
32420750 kg	=	71475519 lb = 35737.8 tons
26598750 kg	=	58640206 lb = 29320.1 tons
18706250 kg	=	41240222 lb = 20620.1 tons
6372500 m	=	20907152 ft
53 cm	=	20.87 in.

Balancing the Shevatim at Har Grizim and Har Eival

In the fall of 1992, there was a fascinating article concerning this week's parsha written up in Tradition magazine by Rabbi Michael Broyde of Atlanta and Steven Weiner of Los Angeles. I will try to sum up the article as concisely as possible. The 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.

Now, in dividing the tribes between two mountains, there are 462 different ways to make such a division [12! / 2(6!6!)]. Broyde and Weiner point out a fascinating fact. Taking the most recent census data that we are given in the Torah and dealing with the answer of the 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.

The next step is what to do with such an impressive observation. What does this tell us? I will leave that for the reader to decide. [In the article, they suggest a parallel to that which we are taught (Kiddushin 40b) that one should always look at the world as if it were half righteous and half guilty and the judgement of the entire world is dependent on him.] But for what it's worth, it is surely an intriguing observation on its own.

The article became the subject of debate in the Spring of 1999 with The Solution to Deuteronomy is not in Numbers, a rebuttal by Sheldon Epstein, Yonah Wilamowsky & Bernard Dickman. This was then followed up by A Mathematical Solution on Terra Firma and a Geographical Explanation on Weak Ground by the original authors.

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.

Splitting up the Animals

Some time after the victorious military campaign against Midyan, ל"א:כ"ה-מ"ז, all of the booty - humans and animals - is counted and divided in two. One half is designated for the soldiers who fought the war and the other half is for the rest of בני ישראל. Of the half that went to the soldiers, one out of 500 was to be given to אלעזר. Of the half that went to the rest of the nation, one out of 50 was given to the לויים.

There are a number of puzzling nuances in this chapter. First the totals of the sheep, cattle, donkeys and humans are tallied. Then the halves to the soldiers are counted as well as אלעזר's portion. The halves to the rest of the nation, although exactly the same as the halves to the soldiers are counted. It is recounted that משה distributed the portion for the לויים but no count is given. Lastly, אלעזר's portion is said to be "from the humans, from the cattle, from the donkeys and from the sheep." The same phrase is repeated with regards to the portion of the לויים but the words מכל הבהמה, from all of the animals, is added.

נצי"ב, in העמק דבר, suggests that מכל הבהמה includes other species of animals that were brought back that were fewer in number. Since they were fewer than 1000, there would not have been enough to give אלעזר even one. Therefore, this phrase is left out of the command of אלעזר's portion and these animals' numbers are not significant enough for the תורה to recount.

A fascinating approach is offered in the name of ר' שלמה הכהן מווילנא. Elazar's portion is referred to in פסוק כ"ט as a 'תרומה לה. One of the laws of תרומה is that one may not separate from one species as תרומה for another. Therefore, אלעזר's portion was required to be one out of every 500 of each animal. However, this was not a requirement with the portion of the לויים and it was sufficient to give them 1/50 of all the animals combined. That is the meaning of מכל הבהמה. The לויים were given 1/50 of all the animals. And that is why the תורה does not go into any detail concerning the division for it was not exact.

Rounded numbers

An interesting article on the use of rounded numbers in the census:

This article was originally found at the following location:

http://www.biu.ac.il/JH/Parasha/eng/bamidbar/mer.html

However, with Bar Ilan's rearranged site, I was unable to locate it. So I rescued the text from the web archive:

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.

According to this commentary, the smallest unit counted consisted of ten (heads of tens), and therefore only groups up to ten were counted and numbers smaller than that were rounded down. This explanation only accounts for part of the difficulty, since almost all the numbers are rounded to hundreds, not tens. Indeed, Rabbi Aharon David Goldberg relates to this question in his book, 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

One of the points of interest concerning the census is the discrepancy between the population of the tribe of Levi as compared to all other tribes. The tally of the tribe of Levi was 22300, almost 10000 short of the lowest tally amongst the other tribes, Menasheh's 32200. But the Leviim were counted from one month old whereas the rest of the nation was counted from 20 years old so their numbers are even more unusually low.

Ramba"n notes this point and offers two explanations: 1) B'nei Yisroel's dramatic increase in population was a result of the subjugation in Mitzrayim. As the pasuk (Shemos 1:12) "But the more they afflicted them, the more they multiplied and the more they spread abroad." Since, as we know, the tribe of Levi was not subjected to the same hardships as the rest of the nation, they did not multiply at the same rate. 2) When Yaakov Avinu expressed his anger with Shimon and Levi over the incident in Shechem, Levi was cursed with being less in number than his brothers.

Ohr HaChayim HaKadosh takes issue with both of these offerings from Ramba"n. First, he argues that B'nei Yisroel's miraculous rate of reproduction was not a result of the subjugation. The pasuk stating, (Shemos 1:7) "And the children of Israel were fruitful, and increased abundantly, and multiplied, and waxed exceeding mighty; and the land was filled with them," comes before any mention of slavery. As far as Ramba"n's second suggestion, Ohr HaChayim cites a census in Divrei HaYamim in which the tribe of Levi was great in number, implying that there was no such curse on Levi.

Ohr HaChayim and Klei Yekar offer an alternative suggestion. The gemara (Sotah 12a) recounts that when Par'oah issued his evil decree on all Israelite males, Amram divorced Yocheved and everyone else followed suit. Although Amram eventually did take Yocheved back, this move had a drastic effect on population growth, and most drastically on his own tribe, Levi. Over 80 years later this was reflected in the census.

R' Sander Goldberg (Baltimore) in Nachal Chayim, shows mathematically how Ramba"n's first answer does not seem to work. B'nei Yisroel totalled 603,550 of which 22,273 were first born. That would mean the first born made up less than 4% of the population. But the first born were also counted from one month. It can be assumed that the total population of B'nei Yisroel counting from one month would be far greater than 603,550. As there is only one first born per family, that means the families had an average size of over 30. This is impossible under natural circumstances and is therefore a testimony to the statement of Chaza"l that the Israelite women would give birth to six babies at a time

When we observe the tribe of Levi we find similar numbers. The population of Levi was 22300 of which 300 were first born. That amounts to even smaller percentage of first born and thus, an even larger average family size! Clearly, when the tribe of Levi multiplied, they did so at a similar if not greater rate than the rest of the nation.

Explaining the Uncounted לויים

This week’s parsha makes it perfectly why this book is commonly referred to in English as Numbers. After counting all of B’nei Yisrael, Moshe is instructed to conduct a ceremonial swap of first-born for Levi’im, a procedure signifying the consecration of the the descendants of Levi as the performers of the service of HaShem, a position previously held by the first-born. First, Moshe counts up all of the Levi’im and the Torah (3:39) reports a total of 22000. The first-born are subsequently counted and their total is 22273. The procedure for the extra 273 does not concern us for now. What is of importance is the point made by Rashi on the tally of the Levi’im. If you add up the figures that the Torah gives us – 7500 for Gershon, 8600 for Kehas and 6200 for Merari – you get a total of 22300!! That would have avoided the need for a special procedure for the extra 273. However, Rashi tells us, based on the gemara (Bechoros 5a) that those 300 extra Levi’im were first-born themselves and therefore, they redeemed themselves, so to speak, and could not be used to redeem other first-born.

Ibn Ezra quotes a complicated calculation from Yehudah HaParsi (whom I believe was a Kaarite,) which he then proceeds to take apart. This is how I, with the help of a friend and the sefer Be’er Yitzchok, understood the give and take in the Ibn Ezra:

Yehudah HaParsi attempts to show how Chazal’s “assumption” that the 300 uncounted Levi’im were in fact first-born is a mathematically sound one. The proposed number of first-born of the Levi’im, three hundred, is approximately 1/73 the size of the general Levite population of 22000. The first-born among the rest of B’nei Yisroel, 22273, were 1/27 the size of the general population. The proportions seem way off at first glance. However, there is one catch. The general population was counted from 20 years old and up. But the first-born were counted from one month and up. Of the Levi’im, however, both the general population and the first-born were counted from one month.

Yehudah HaParsi proposes the following adjustment: Beginning at the end of this week’s parsha and spilling over into next week’s, the Levi’im of the age of service are counted. The total given (4:48) is 8580. Subtracting the 300 first-born, we are left with 8280. The Levi’im of the age of service therefore make up a mere 38% of the total Levite population (8280/22000=0.38). If we were to take only that percentage of the first-born of the rest of B’nei Yisroel, there would be only 8383 first-born of the age of service ((8280/22000)*22273=8383). This is remarkably 1/73 of the general population of B’nei Yisroel which was initially tallied based on service age, an astonishingly accurate correlation with the Levite figure of 1/73. This is truly a brilliant calculation.

However, Ibn Ezra didn’t think so. He strikes down the entire calculation with one very simple fact that I deliberately avoided exposing until now. The age of service for the Levi’im was from 30 to 50. The counting of B’nei Yisroel began at 20 years old without any upper bound. Thus, there is no rationale for comparing the two figures. [There are other mathematical flaws as well. It is foolish to subtract all 300 first-born Levi’im from 8580. Either the first-born should be subtracted proportionately (117) or the 8580 should simply be divided by 22300, ultimately resulting in 1/71 as the proportion of regular first-born.] Rather than trying to come up for some “proof” for the validity of the words of Chazal, we must accept them as truth with full faith that that is what was passed on to them.

What are the odds?

Since this week's parsha deals at length with first-borns, I thought I'd share a rather interesting family fact:

I have an aunt and uncle who have six children (בלי עין הרע) and every single one of those children made a פדיון הבן!

I once tried to calculate the odds of that happening. To calculate the odds of anyone making a פדיון הבן there are a number of factors that must be calculated. We can try to approximate:

1. 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.)
2. The first fetus has to be male (let's just say 50%)
3. The baby must be delivered and not miscarried (let's use 90%)
4. The baby must be born without a Cesarean (again, 90%)

So the odds of any non-Levite making a פדיון הבן are only about 32%. The odds of going 6-for-6 are a mere 0.12%

WOW!

The Probability of the Goral

In 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.

Even though the portions were Divinely predetermined, a lot-drawing process was used to assign each tribe their portion. Rashi explains that one drum was filled with 24 pieces of paper. On 12 pieces of paper were written the names of the 12 tribes. On the other 12 were written the 12 portions that were to be assigned to the tribes. Each Nasi approached the drum and picked out two pieces of paper. One paper had the name of his tribe written on it and the other the prescribed portion of land in Eretz Yisrael. The purpose of this exercise was to prove the Divinity of division plan that allotted each tribe its portion and appease any tribe who felt it might be unfair. As such, I believe that a miracle of this type may be more greatly appreciated if we knew exactly how unlikely it would have been to happen naturally.

Suppose we had a prescribed list of which portion was to be assigned to which tribe. What would be the odds of each Nasi picking out both the name of his tribe and also the corresponding piece of land that had already been prescribed? Let us start with the first Nasi. He has 24 pieces of paper to choose from. He must pick two specific pieces of paper out of the drum. The odds of taking the first one correctly would be 1/24 and then the odds of taking the second correctly would be 1/23. However, since the two papers were taken together, the order does not matter. The rules of probability theory state that if the order of the choices is not relevant, than the odds must be multiplied by the number of possible sequences which, in this case, is two. So the odds of the first Nasi picking the right pieces is 2/(24 x 23). With 22 pieces of paper remaining, the odds of the second Nasi picking correctly will be 2/(22 x 21). And so on. The last Nasi's odds will be 2/(2 x 1) which is 1. That means that he will definitely pick the right ones. That is understandable. By the rules of probability theory, in order to find the odds of all the Nesiim picking correctly, we must multiply each Nasi's odds together. Thus, the odds may be generalized as

212

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

24 x 23 x 22 x 21....x 2 x 1 is referred to as 24 factorial and expressed as 24! Thus the final expression is 2¹²/24!. 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.

This calculation is based on Rashi's explanation in the chumash according to the Midrash Tanchuma. However, Rashi (actually written by Rashbam, Rashi's grandson,) in the gemara (Bava Basra 122a) states clearly that two drums were used. This will alter the calculation somewhat. The first Nasi would have a 1/12 chance of picking his tribe's name from the tribe drum and a 1/12 chance of picking the correct portion from the portion drum. The fact that the order does not matter will not affect the odds in this case because the choices are made from two separate groups. The odds for both drums are multiplied and thus, the first Nasi's odds will be 1/12². The odds of the second Nasi will be 1/11². And so on. The total odds will be:

1

122 x 112 x 102 x 92 x 82 x 72 x 62 x 52 x 42 x 32 x 22 x 12

This can, in fact, be simplified as 1/12!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.

Just to get an idea of the extent of this improbability, the odds of winning the Powerball Jackpot are approximately one in 175 million. That is over 1.3 billion times more likely to happen than this, according to Rashi on the gemara, and over 860 billion times more likely according to Rashi on the chumash. The odds of getting fatally hit by lightning in a given year are approximately 1 in 2.4 million. In fact, it is more likely for one to win the Powerball twice in one week or get fatally hit by lightning two-and-a-half times in one year than for the 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.

Counting the Judges

In the end of Parshas Balak, (pasuk 25:5), Moshe passes on HaShem's command to carry out justice upon those who worshipped Ba'al Pe'or.

Rashi states that there were 88,000 "Dayanei Yisroel" and cites the gemara at the end of the first perek of Sanhedrin. However, the gemara over there clearly calculates the number of Dayanei Yisroel to be 78,600¹. 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)

A population of close to 672,000 is needed to necessitate 88,000 dayanim. This is an odd number since none of the recorded censuses rendered a number anywhere near this. But as clearly shown in the very Rashi in question, there was to be a large population decrease before B'nei Yisroel would reach the figure of 601,730 given in Pinchas. To justify this figure of 671,755 we must account for 70,000 lost lives. The only definite casualty count we are given is the 24,000 who perished in the plague following the worship of Ba'al Pe'or. That still leaves 46,00 lives unaccounted for. Starting from Beha'aloscha there were a number of catastrophic incidents recorded in which many fell from B'nei B'nei Yisroel. However, many of these may not be considered in this particular calculation. If B'nei Yisroel did in fact reach a population as large as we are suggesting then it must have happened gradually from the time of the census in Bemidbar to the time that they began their decline to the figure given in Pinchas. Therefore, since the individual plagues from Beha'aloscha to Korach were still in the second year and, for the most part, immediately after the census in Bemidbar we may not consider them in the decline of the population toward the figure of 601,730.

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 casualties².

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

There are a number of details involved in this figure that may be questionable. Firstly, we have previously determined that before the sin of Ba'al Pe'or the population was at least 625,000. Also, there is no source that indicates that all these different parties were included in the term "Dayanei Yisroel." The gemara in Sanhedrin certainly did not include them. Then Rashi's comment on this pasuk would have absolutely nothing to do with the gemara and from the text of Rashi it seems that Rashi himself cited the gemara in Sanhedrin. So, the conclusion remains that the proper reading of Rashi is most likely 78,000 rather than 88,000.

¹Although 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.

²The 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)