Happy π Day

We wish you all a happy Pi Day, today being March 14th which, in the US anyway, is expressed as 3-14. Pi day was first observed in the year 1593. Ok, I'm just making that up (and rounding.) Just to give this some semblance of a Torah flavour, here is our post on Pi in the Torah 

In European countries where the day is written before the month, Pi Day is observed on April 31. For information on that, you would have needed to contact me this past Sunday morning at 2:30 am.

והמבין יבין.

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

עמודי החצר

In the end of Parshas Terumah, (pesukim 27:9-19), the Torah describes the beams that held up the curtain that surrounded the courtyard of the Mishkan. Pasuk 10 discusses the beams on the southern side of the courtyard:

In Rashi's seemingly innocent comment on the pasuk, there is a grave arithmetic difficulty which is the subject of much discussion amongst the commentators on Rashi. If there are five 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.

Firstly, the Mizrachi suggests that the five 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.

In pasuk 18, the Mizrachi suggests that the 100 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.

The Levush HaOrah, another commentator on Rashi is unhappy with both the Mizrachi and the Gur Aryeh's explanations of Rashi in regards to the placement of the beams. From the fact that Rashi mentions the measurement of five 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.

Finally, the sefer Ma'ase Choshev offers another possible arrangement of the beams which matches that of the Levush's in symmetry and arithmetic correctness. He suggests that there were no beams in the corners. The curtains were suspended from wooden bars. On these bars were placed the hooks that were used to hang the curtains from the beams. Each of these bars was five 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 amosrefers 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.)

The arrangement of the Ma'ase Choshev is the one quoted in the seforim Meleches HaMishkan and Tavnis HaMishkan (etc.). The sefer Lifshuto Shel Rashi, however, is content with the opinion of the Riva and the Abarbanel. Whatever the true arrangement of the beams was, it is clear that when Rashi said that there were five 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.

חמושים

When B'nei Yisrael leave Mitzrayim, the pasuk says (13:18) that they left "chamushim." Rashi says that the literal meaning is that they left armed. But he brings a Midrash from Mechilta that only 1/5 of B'nei Yisroel came out of Eretz Mitzrayim and the rest died during the plague of darkness because they did not want to leave. In the Mechilta itself there are two other opinions, that 1/50 or 1/500 of B'nei Yisroel came out. This is indeed a disturbing Midrash. It would mean that 2.4 million, or 24 million, or 240 million of B'nei Yisrael died during the plague of darkness. That would make it a far greater blow than everything brought upon the Mitzrim put together. R' Shimon Schwab in Ma'ayan Beis HaSho'eva asks, as well, that the point of the B'nei Yisrael dying during the darkness was so that the Egyptians would not see them dying. It's kind of hard not to notice at least 2.4 million people missing. 

 R' Schwab answers, therefore, that really only a few people died at that time. The discussion in the Midrash is concerning the long term effects of it and how to gauge it. The first opinion looks only so many generations down the line and sees that at that point, 2.4 million Jews that would have been born never were. The other two opinions simply look farther down the time line to see the greater long term effects of this loss. This would answer all the questions. But what I find very bothersome about this answer is that the wording of the Midrash simply does not seem to lend itself to such interpretation. The number of deaths is not given at all but rather the fraction of B'nei Yisroel that left Eretz Mitzrayim. This fraction would not change over the generations. In other words, let's say 600,000 went out but it could have been 3 million. If generations down the line, we reached a population of 6 million, the projection should dictate that we 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.

Can you Count to 70?

Question: How many males are counted as coming to Mitzrayim with Yaakov? One thing is for sure, it wasn't 70. I still have not been able to figure out how all the numbers worked - who were the 66 mentioned in 46:26 and the 70 in 46:27? 66+3 = 69, the last time I checked. If you add up all of the children and grandchildren, it does come out to 70 but then it should have been 67 and then 70. All that aside, it was not only males who were counted. Dinah is counted along with her brothers which is understandable. Serach bas Asher is counted as well which is slightly more puzzling. One must assume she was not the only granddaughter. From Rashi (46:27) it seems Yocheved was somehow part of the 70 as well.

While I was not able to find anything explaining why these particular women figured in the count as opposed to others, I did see an interesting insight into the pesukim in consideration of that fact. Tzeror HaMor and Emes L'Yaakov both point out a discrepency in the per-wife tallies found in the pesukim. The numbers for Rachel ("arba'ah asar") and Bilhah ("shiv'ah") are of the masculin form. The numbers for Leah ("sheloshim veshalosh") and Zilpah ("sheish esreih") are feminine. They both explain that Leah and Zilpah both had women counted among their offspring - Dinah from Leah and Serach from Zilpah. Therefore their numbers are delivered in feminine. Rachel and Bilhah had no feminine offspring counted and thus their numbers are in masculine.

One might wonder why this is so, considering that the generic plural is usually masculine by default. However, Emes L'Yaakov points out that the word "nefesh" which the number is qualifying is feminine. So the default gender of the number for "nefesh" should be feminine. Rachel and Bilhah were the exceptions.

Goats and Amicable Numbers

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)

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.

Moshe's pleas

At the beginning of this week's parsha, Moshe mentions that he pleaded with HaShem to allow him to enter Eretz Yisroel but to no avail. The sefer M’galeh Amukos says that Moshe Rabbeinu davened 515 times - the gematria of Vaeschanan. R' Yehonasan Eybeschutz, in Divrei Yehonasan, is curious to discover how such a tally is reached.

He offers the following possibility: The Midrash states that Moshe Rabeinu started davening on 15 Av. As the gemara (Bava Basra 121a) explains, it was on this day that it was realized that the punishment for the sin of the spies was complete and no more men would die in the midbar. He saw that that decree had been fulfilled and had a glimmer of hope that perhaps, since he had been spared from the decree, he was in a position to plead for Divine Mercy. (This would explain why he never engaged in such extensive prayer on Aharon's behalf as Aharon died prior to 15 Av.)

There are 6 months from Elul to Shevat. We may assume that it was a normal year, whose months alternate between 29 and 30 days throughout. So those full months would total 177 days (3x30 + 3x29). Add the 16 days of Av that Moshe davened and the 7 days of Adar until he dies and we have 200 days. Of those 200 days, 28 are Shabbosos on which it is not permissible to make personal requests. That leaves 172 days. Considering Shacharis, Mincha and Maariv and we now have 172x3 = 516 tefilos. Only off by 1. However, the nation only discovered in the morning of the 15th of Av that the dying has stopped. Therefore, Moshe would have missed the Maariv from the night before and only begun davening at Shacharis. And there you have exactly 515 tefilos!

The Tur writes that on Yom Kippur one is permitted to make personal requests, but on Rosh HaShanah, Sukkos, or Shmini Atzeres it is forbidden. We would then have to subtract three more days of prayer. However, we are taught that Moshe Rabbeinu died on Shabbos. If that is the case, then Rosh HaShanah, Sukkos, and Shmini Atzeres of that year all fell on Shabbos as well. So we need not subtract for them and we are safe with our tally of 515!

Special thanks go out to R' Ari Storch for providing me with the material for this shtikle.

Gematrias off by 1

One of the favourite, and often entertaining forms of drashos is the 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.

The reading for the morning, taken from this week's 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.

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.

No Population Increase

I was discussing the census numbers with someone one שבת. An interesting question was posed regarding the lack of an increase in population over the different censuses that were taken throughout the years in the מדבר. This question is really better suited for פרשת פינחס which takes place towards the end of their journey with still no increase. 

One obvious question is that there should have been many children born over the course of the first 20 years in the מדבר who would be counted by the last census. I've heard some answers to that question which I'd rather not go into at this juncture. The less obvious but more difficult issue is the children that came out of מצרים. As we've pointed out in a previous post, the first born made up approximately 4% of the population which means each family was exceedingly large. It would probably be a gross understatement to suggest that each family consisted of at least 10 male children. Let's even go so far as to say 5, to take into account children who were already counted in the original census. Even though the original 600,000 included a number of different generations it still seems that by all accounts, there should have already been millions of male children not counted in the first census. So where did all these millions go?