The numbers in the Book of Numbers, especially those of the socalled census, belong to the most mysterious parts of the Torah. For more than two milleniums people have tried to find out what the real meaning of these exceedingly high quantities could possibly be. The first and simplest explanation was to take them literally as they stand. For instance Paul in his first epistle to the Corinthians 10,8 argued: "We should not commit sexual immorality, as some of them did  and in one day twentythree thousand of them died.", apparently hinting to Numbers 25,19 . However, there is a consensus among most exegetes today that the numbers are much too great to be taken literally but should be interpreted in some other way: symbolically, astronomically/ astrologically, as mere random numbers without any historical value, historically but distorted by mistakes during textual tradition, and so on. You will find important contributions in my literature list. One of these contributions (by Prof. Humphreys) has already been discussed in the review section of my homepage. Basic information of 'Akzent 7' can be found in my article "Bileams Rätsel  Die Zählung der Wehrfähigen in Numeri 1 und 26", which I wrote in 1996 and which in turn is based on observations I made in September 1995. 'Akzent 7' is not a simple translation of that paper but you can get all the relevant ideas from reading this 'Akzent'. So here we go: The number collections in Numbers 1 (  4 ) and 26 (see table 1) constitute quantitative structures (QS) in a narrow sense as defined in 'Akzent 1'. When I analysed these structures closely I discovered a couple of statistical peculiarities giving a considerable amount of information for a 'decoding' of these numbers. 
Tribe 


Reuben 


Simeon 


Gad 


Judah 


Issachar 


Zebulun 


Ephraim 


Manasseh 


Benjamin 


Dan 


Asher 


Naphtali 





Table 1 : The numbers of the censuses according to Numbers 1 and 26
Concerning the motivation for such a decoding we must see at least
two different things. On the one hand  as mentioned above  the sheer
greatness of the numbers leads us to ask for a nonliteral meaning. On
the other hand, the interpretation of this fact as presented
in the Book of Numbers gives additional reason to ask for an alternative
explanation. For example, there is the
dazzling figure of Yahwes prophet Balaam who has not been sufficiently
explained, yet. It is the Moabites fear of the greatness of
Israel by which Balaam's appearence in the middle of the book is being
motivated (Num.22,26). In his first oracle the prophet notices: "Who can
count the dust of Jacob or number the fourth part of Israel?" (23,10a).
This rhetorical question is generally regarded to be a poetical description of the enormous greatness of Israel  as we can see it already in the blessing of Abraham (Gen.13,16). However, strictly speaking it stands in contradiction to chapters 1 and 26, because here  at the behest of the very God (Num.23,5a, Gen.13,14a)  this impossible thing is being carried out even twice. Balaam as a prophet is speaking in riddles  unlike Moses (12,68) [1]. His oracle can be 
read as an insinuation
that those 24 numbers and their sums are not a census result. Provided
the writer of the Balaam story knew about the artificial origin of the
numbers in chapters 1 and 26, then Balaams oracle can be a deliberate hint
to this fact [2].
For this reason  but of course not for this reason alone  it is a conundrum how the numbers of this 'impossible' census are to be explained. However, without having an explanation of these numbers it seems to be impossible to really understand the Book of Numbers or the Moabites fear of the greatness of Israel. In this 'Akzent' we will follow up Balaam's hint using further contextual and statistical observations. The interpretations presented so far rarely treat the 24 numbers otherwise than they seem to be intended in their immediate context [3], disregarding that many commentators believe them to be simply invented (which does not explain why exactly these numbers were invented). Finding that at least Enoch's age among those of the patriarchs in Gen.5 can be explained in a completely different way than the context suggests (cp. 'Akzent 6'), it appears to be rational to apply a similarly radical procedure in Numbers 1 and 26 as well. 
Looking more closely at table 1 we discover some striking features: a) Only the numbers 45,650 for Gad in Num.1 and 43,730 for Reuben in Num.26 have no zero in the second lowest decimal place [4] and so are given a little bit more precise than the others. In connection with these two numbers we observe the following in the wider context: In Num.3,4051 we read about the counting of all firstborn Israelite males (22,273) as well as of a strange [5] compensation for these firstborn by the Levites (22,000). It yields a surplus of 273 men. For each man 5 shekels are to be given to Aaron and his sons, in all 1365 shekels. 273 (and 22,273 resp.) is probably the least understood number in the Book of Numbers. Now there is a correspondence between 273 and 1365 on the one hand and 43,730 and 45,650 on the other hand, in that the last two nonzero decimals are equal to 73 and 65 in both cases. For this reason the numbers 273 and 1365 are assumed to have been layed down as key numbers for the reconstruction of an original set of numbers. The following fact can be used to apply this key: b) The last digit of all 24 numbers is zero. It has often been suggested to delete at least one zero in all numbers since they are so unrealistically high. In doing so, the two slightly more precise numbers of a) are changed into 4373 and 4565. Then 73 and 65 stand in the same decimal place as in 273 and 1365. The latter can e.g. easily be subtracted from 4373 and 4565 resulting in especially 'smooth' numbers. 
c) There are two double numbers:
40,500 for Ephraim in Num.1 is repeated in Num.26 for Gad. 53,400 for Naphtali
in Num.1 is repeated in Num.26 for Asher. However,
the probability for two double numbers to occur is rather small [6].
d) Furthermore, comparing the hundred digits of the numbers in chapters 1 and 26 we get the impression that they originate from the same distribution, because they occur with very similar frequencies: Zero: (0,0), One: (0,0), Two: (1,1), Three: (1,1), Four: (4,3), Five: (3,4), Six: (2,1), Seven: (1,2), Eight: (0,0), Nine: (0,0) [Format: "decimal digit: (frequency in Num.1, frequency in Num.26)" ]. In order to explain d) and c) it is assumed that in both of the original numerical series the frequency of digits was completely balanced. After that the two double numbers were 
placed in the least
conspicuous way, i.e. in the other census, respectively. To decode the
numbers we can delete two of them so that in both columns the frequency
of digits in this decimal place becomes equal.
Motivated by these observations we now apply three transformations: 1. Division by 10 (deleting the trailing zero in all 24 numbers). 2. Substitution of 4565 by 3200 = 4565  1365 in Num.1 and of 4373 by 4100 = 4373  273 in Num.26. 3. Deleting 5340 in Num.1 and 4050 in Num.26. We get the 22 numbers and two sums in table 2. 


























The names of the tribes have been omitted because they
cannot be uniquely related to the numbers. There is nothing special about
the sums in the last row. 
In table 2 the decimal digits in the tens place (before: hundreds place) now occur with exactly the same frequency. Going from 0 to 9 we get the following picture: (1,1), (0,0), (1,1), (1,1), (3,3), (3,3), (1,1), (1,1), (0,0), (0,0) [format as above]. So it presents itself to combine the numbers 'horizontally' as pairs according to these decimal digits. However, since there are 3 fours and 3 fives the creating of pairs is not unique. e) Now in table 2 we find a rather similar coincidence of digit frequencies in the hundreds place: (1,1), (1,1), (3,3), (0,1), (2,2), (1,2), (1,1), (1,0), (0,0), (1,0). Analogous to the tens place we can suppose that a couple of digits in this decimal place as well did not accidently coincide with their partners. f) Furthermore, with regard to the tens and hundreds place we notice pairs of numbers having the following digit combinations: 05, 22, 27, 44, 54 und 65 [7]. It is not that astounding to find double or multiple pairs of digits in a sample of this size, but that there are exactly 6 double pairs. (In this number the doubling of whole numbers as discussed in c) is not included.) So for 12 numbers there is a close connection between the digits in the tens and the hundreds place [8]. Therefore we are certainly entitled to use the hundreds, too, in order to build unique pairs in those cases where a four or five is given in the tens place. 
As next step we apply a corresponding transformation:
4. Building of pairs that are equal in the tens place. Make use of the hundreds place where the tens place does not lead to a unique relation. (See table 3 left part. For clearness the pairs have been ordered vertically according to the decimal places, respectively. This is of no influence on the result.) This procedure created pairs of numbers in each row that have equal digits for all tens and for six out of eleven hundreds. g) Let us now compare the six multiplied by two numbers with equal tens and hundreds concerning their positions in Num.1 and 26. It is astounding to find one number of each pair in chapter 1 and the other in chapter 26, isn't it? 
And all of these numbers are directly taken from the
Bible text, i.e. they were not changed by any of the transformations (except
the deleting of zero). So not only their creation
cannot be reduced to chance but also their symmetrical distribution in
Num.1 and 26 has to be explained [9].
It suggests itself to assume that the numbers of both chapters were taken from the identical distribution, perhaps even developed from each other. In order to find out a possible system we now relate the numbers to each other horizontally applying a last transformation by subtracting them, similar to the application of the key numbers above (transf. 2). The pairs are turned round where negative differences occur. 5. Subtract the individual numbers of each horizontal pair. For negative differences the pair  now called minuend and subtrahend  is being turn round. (See table 3 right part. The turned pairs are marked by () in the 'difference' column.) 











() 900






1000








() 500







() 1000





() 1000






400






() 2000






900






() 3000






2900






1000


Sums:





14600


=
146,986...
* 365 
=
153,013...
* 365 
= 170 * 365  = 130 * 365 
= 40 * 365

By the last transformation the sums became dividable by 365. Together
they form the difference
The numbers 170 and 130 contain the factors 17 and 13 which are known to be gematries of God's name Yahwe (17) and the numeral 'æhad (13). Taken together these words can mean 'Yahwe is One' (cp. Dtn. 6,4) [11]. We can say: the 40 year desert journey is under the banner of Yahwe the one and only God of Israel. 
Practically every decoding is way more difficult to do (without knowing the encoding method) than the encoding procedure. For this reason the five transformations seem to be complicated at first glance. However, they do not use any 'advanced' mathematics but only shifting (transf. 1), a key (transf. 2), doubling/deleting (transf. 3), addition/subtraction and two simple rearrangements. These procedures belong to the simple part of the inventory of mathematical methods that were already at hand in ancient times (cp. Young 1 / 2). Their application to encoding is trivial. But it has to be asked if the people in ancient Israel would have been able to retrieve this meaning from the text by decoding it without knowing the encoding method. Even for the educated of their time this can scarcely be supposed. I assume this design to have been understood by the designer himself as well as some of his students, but that it sank into oblivion in course of time, whatever the circumstances may have been. The challenge remained to understand the riddle of these numbers down to its details. Concerning the individual transformations: The increasing of all numbers by a common factor (transf. 1) is well explained by Davies' proposal, that Yahwe's miraculous power in keeping up such a great people ought to be delineated. According to the oracles of Balaam it is a matter of strength and invincibility of the people (23,22.23 24,9) as well as entitlement to domination and landed property (23,24 24,7b.8.14.1719). At the same time Balaam is speaking with constant regard of the nearly unimaginable greatness of the people: At first he is allowed to see the outermost part only (22,41 23,13), changing his point of view should help him in cursing (23,13.14.27.28). Finally he is overwhelmed by what he is 'seeing' (24,1.2.5.6). 
After transformation 2 all numbers again have a zero as last decimal,
which suggests to assume they had only three digits when they were selected. The
occurence of familiar numbers from astronomy, like 354 (length of a lunar
year) and 40 * 365 (40 solar years) as well as 465 and 765 (with additive
component 365 ?) calls attention to an interest in such numbers [12].
The key numbers 273 and 1365, too, point to astronomical periods. 1365
clearly contains 365, 273 is 3/4 of the 364day year which is known from
the Book of Enoch as well as the Book of Jubilees [13].
(However, 273 could as well be a rounded 3/4 of the 365day year.) Perhaps
the designer thought these features to be especially clear hints to the
symbolical origin of the numbers in the Book of Numbers.
The striking frequency of numbers with equal decimals suggests that in constructing the 
sums great rounded numbers were added to well known symbolic numbers. It is possible e.g. that 454 = 100 + 354 or 222 = 200 + 22 (22 being the number of letters of the Hebrew alphabet) and so on. If this was the method of construction then it is doubtful if it is possible to determine for each individual number how it was made, because such additions are not unique. However, this method makes clear why the numbers in chapters 1 and 26 come from the same distribution. The presence of several free parameters is disturbing when reconstructing, but they are an advantage for design, perhaps are even necessary, because it should be difficult to enforce the resulting 170 * 365  130 * 365 = 40 * 365 by using symbolic numbers only. So if it should turn out to be impossible to make clear where the individual numbers come from, this does not tell against the solution presented in sections 2 and 3. 
The key numbers, too, hint to the additive composition
of the individual numbers and the multiplicative composition of the final
result. They can be written as: (1000 + 365) = 5 * (200 + 73). Seen this
way they contain the means of construction in essence, the statement being:
By adding and multipliying of symbolical and rounded numbers new numbers
conveying symbolic meaning can be made. 
Reducing the two number columns from 12 to 11 members each (transf. 3) prevents a unique relation to the 12 tribes of Israel as they are counted in the Book of Numbers. It looks like as if the designer of this construction had a different imagination of Israel at first and only later created the current relation to the 12 tribes by doubling the 40,500 and 53,400. One could assume e.g. that the original approach set the tribe of Levi against the remaining 11 tribes of Israel before the partition of the tribe of Joseph. However, perhaps the doubling of two numbers was only meant to be a hint for a later reconstruction.  The transformations 4 and 5 can be carried out differently. The sortings that were employed are near at hand because of the balanced distribution of digits. The horizontal changes have altered the reference point of some numbers with regard to beginning and end of the desert journey. In this case as well the relation would not have been defined in the first step but determined only later. It may have emphasized the tribes according to theological viewpoints.  
A definitive interpretation of the individual minuends, subtrahends and differences has yet to come. However, since these three numbers are connected by an equation, resp., a special selection of each of them would result in an 'over determination' [translation?]. So we can expect a special determination only for part of the numbers. But it is improbable that any of these numbers were 'simply invented', as some have assumed. The statistical observations of section 2 already demand a systematic explanation. Following a proposal of Claus Schedl interpreting the patriarchal ages in Gen.5 , one could look for text sections that correspond to the 11 differences, if they are measured in numbers of words. This would be a technicalfunctional possibility of interpretation, besides the number symbolical i.e. gematrical (17 and 13) and time period related (40 and 365) explanations. However, it  is not quite certain from the beginning that each of
the 11 differences has a meaning of its own (they are not written in the
text). In any case, the precise greatness of the individual numbers remains
to be explained. 
Finally it has to be asked: What is the sense of such an encoding in the Torah? The answer is: The contents of the book ought to be deepened. The author of the numbers wanted to glorify his God openly (by the greatness of the numbers) and in concealment (by encoding his credo). Here it is surprising at first how sovereignly he proceeded. He sets aside the 'historical truth' of the numbers and instead implants them his theological message of Yahwe, the One and Only also during the wilderness journey. However, his souvereignty corresponds very well with the souvereignty of the author of the Torah, the lawmaker himself. 
A modern counterpart to this number art of words can be found in the number art of music by Johann Sebastian Bach. As we all know he designed many of his works with respect to symbolic numbers. In his 'Musicalisches Opfer' ('Musical Sacrifice') which he dedicated to Friedrich the Great in 1747 there are some riddles embodied. On the presentation page the king is requested to look for (ricercare) and to solve them. (To some extent this hint corresponds to Balaam's enigmatic oracle.) However, finally the open as well as the hidden constructions in Bach's work ought to serve God's honour alone. The Torah numbers of the One and Only equal the abbreviation S.D.G. in the scores of Bach: Soli Deo Gloria! In both cases the will of a superior master manifests itself, to designate the object that causes the unity of his thinking and living. 
How good is the evidence for the validity of the interpretation given in section 2, that the structure 170 * 365  130 * 365 = 40 * 365 has not been caused by chance? The probability p for an accidental occurrence of the structure is given by the sum of probabilities for all paths of transformation that lead to this or to a similar meaningful result. If there were no striking statistical observations then there would be hardly any appreciable limits for the number of alternatives and the only senseful estimation for p would be 100%. However, in course of the decoding a couple of statistical phenomina were taken into account which have to be explained by all other possible transformations as well, especially: (a) the double numbers 40,500 and 53,400 , (b) in Num.1 and 26 comparable frequencies of digits in the least significant decimal place , (c) six pairs of numbers whose constituents have two digits in common, and (d) the symmetrical distribution of these numbers among Num.1 and 26. (These points relate directly to the biblical text and do not result from the transformations presented in section 2.) These four observations suggest a transformation of the numbers as presented in this article. It can not be completely excluded that principally different interpretations may be found. (These, too, would have to explain points (a) to (d). ) However, there do not seem to be many alternatives of this kind, because two millenia of Numbers  exegesis have failed to come up with a single satisfactory approach, and we can rest assured that a very great number of interpreters have tried to solve this problem. 
So the probability for principally different explanations is very
small, but it cannot be quantified. For this reason also the over all probability
p cannot be estimated in a quantitative way, i.e. it is not possible to
describe the probability for the accidentical occurence of the structure
by
a single number.
But it can be seen that the five transformations are an optimum with regard to several viewpoints, especially (a) to (d). Beginning with the last transformation: If we alter the horizontal changes of the six minuends and subtrahends in one way or the other (there are 64 possibilities) then in any case the result 170 * 365  130 * 365 = 40 * 365 vanishes and we get individual differences with negative sign. The same holds when arbitrarily changing the 11 pairs (2048 possibilities), disregarding the case where all signs are changed, opposite to the presented solution for symmetrical reasons. Above all the distribution of different signs of the individual difference would then need to be explained (let alone the question how to interpret the deviating numbers of the final result.) The need for explanation arises in 
view of the assumed method of design (cp. section 4):
If the 11 numbers of one of the columns emerged from the other column by
adding round numbers, then the differences should have the same sign. The
current solution has no problem with that.
The different signs can be avoided by questioning also the building of pairs in transformation 4. However, in this case it has to be specified why the statistical observations (b), (c) and (d) are being disregarded and building of unique pairs according to these criteria is declined. No matter how the reasons for this would look like, still the alternative then chosen would have to explain the points (b), (c) and (d)! But there is only one possibility to build unique pairs according to (b), (c) and (d) and to achieve uniformity of signs, and this is the solution presented here. Transformations 4 and 5 presume the simple idea that the numbers were originally conceived homogenously in regard to sign and value and were only later alotted to the tribes according to unknown (theological?) viewpoints, and thereby mixed up. 
It was mentioned above that the 11 differences are not
written in the bible text and that it is uncertain in this respect if they
have a meaning of their own. By this approach the occurrence of different
signs looses its significance and the question arises how often the result
170 * 365  130 * 365 = 40 * 365 occurs at all, given the possible
combinations of the 22 numbers. The chances to get this result (or the
one with changed signs: 130 * 365  170 * 365 =  40 * 365) by selecting
any combination of 11 out of the 22 numbers are 1 : 4100. I.e. seen
as
such the result of transformations 4 and 5 is significant anyway.
Let us now consider the first three transformations: The deleting of zero (transf. 1) is a step that can presumably be accepted most easily and is of little influence on the problem of evidence. It will not be further discussed here. Now the subtraction of the key (transf. 2) 'creates' zeros in the final two digits of the original numbers throughout, so wasn't it aimed at promoting the formation of a meaningful sum or difference? In fact, for the purpose of decoding a simplification was aimed at. However, it does not increase the number of potentially meaningful series of digits, but it decreases it. This is because for all possible sums transformation 2 forces the digits of one more decimal place to be zero, i.e. possibly existing information is annihilated. But it is just this data reduction which leads to the solution. 
The numbers 1365 and 273 being a key for the reconstruction of the
original numbers, this supposition cannot be realized as correct by considering
transformation 2 by itself. Their interpretation as key appears to be plausible
when we see their exceptional position in the wider context as well as
the
coincidence of the digit pairs 73 and 65 with the digits of exactly those
numbers, which are given slightly more precise then the others, finally
criterion (c) which signalizes an extraordinary relevance of double pairs
of digits.
Above all it is the improbability of double numbers to be created by chance, criterion (a), that speaks for transformation 3 (deleting one of the two double numbers, resp.). A doubling by hand of man seems to be the most simple assumption, given the shortage of alternative explanations (disregarding chance, there is none). In particular this explanation makes sense within the framework of the reconstruction as put forward here (cp. section 4). Summing up, for the first three transformations it may be said that eventually their success speaks for them: They enable a conversion of the striking statistical peculiarities (b), (c) and (d) into two additional 
consequent transformations that finally show the numbers
to be surprisingly fraught with meaning. 
Beyond statistical analysis the following arguments can be put forward in favor of the validity of the result: One main advantage of the decoding presented here is the fact that the numbers in their entirety have a special meaning. Earlier attempts at explanation were mostly restricted to gematrical or astronomical interpretations of single numbers. A second important merit is the astoundingly clear hint (40 * 365) to the desert journey, the beginning and ending of which is marked by the censuses. Thirdly, the reference to Yahwe as the one and only God of Israel puts the grandiose image of the wilderness journey  including its basic item: Yahwe's self revelation to his people  into the highest theological context. Finally, Balaam's enigmatic oracle is answered: Who can count Israel? (Little surprising) the answer is: Nobody. When interpreting the censuses in chapters 1 and 26 as put forward here, then Balaam's statement is consistent within the Book of Numbers as well as beyond. 
(Akzent 7: The Numbers In The Book Of Numbers) Balaams utterance concerning the countlessness of Israel gives reason to look for an alternative explanation of the 'census of the people' in Num. 1 and 26. Using a key layed down in the context and a number of statistical observations, the figures can be decoded yielding the formula 170 * 365  130 * 365 = 40 * 365 . These numbers can be interpreted within the meaning of the gematries of Gods name Yahwe (17) and the numeral 'æhad (13) (cp. Dtn. 6,4) as well as the reading of 40 * 365 as the 40 years of wandering in the desert: Yahwe, the one and only God of Israel, presided over the 40 years of their wilderness journey. 
End of 'Akzent 7'
Click on a keyword or number to go back to the text.
[1]
: He has to say unrealistic things on other occasions as well: Num.
23,21a.  Cp. also Num.24,18 and Dtn.2,5/Num.20,19.
[2]
: This corresponds to Yahwe's promise to Abraham.
[3]
: There are some short remarks on these in my German article footnote
6.
[4]
: Michel Barnouin (p. 285)
calculates: 45,650  3550 = 42,100 (Gad in Num.1) and 43,730  1730 = 42,000
(Reuben in Num.26) , where 3550 and 1730 are those parts of the sums, that
exceed 600,000, respectively. This way he succeeds in getting two sums
of exactly 600,000 , which is interesting, because there is a similar dissection
in Ex.38,2528 (pp. 281/282).
[5]
: Disregarding the theological aspects it is strange, because an
apparently rounded number (22,000) is being compared with a number three
times as precise (22,273).
[6]
: Supposed, the creation of the 24 numbers could be approximated
by a normal distribution, then the probability for the occurrence of two
double numbers is approx. 2.3 * 10 ^{ }^{4}
.
[7]
: The numbers are: 4050/6050, 3220/2220, 6270/5270, 5440/6440, 3540/4540,
4650/7650.
[8]
: This observation justifies the attention given to the digit pairs
73 and 65 in the key numbers.
[9]
: The probability for an accidental symmetrical distribution is approx.
10
^{ 3} .
[10] :
40 is a round number. Cp. e.g. . Ex.16,35, Num.14,34, 32,813, Dtn.2,7
and 14.
[11] :
y + h + w + h = 1 + 5 + 6 + 5 = 17 according to the socalled 'small
counting', when 10 (y) is counted as unity. ' + h + d = 1 + 8 + 4 = 13.
 Cp. Labuschagne
(pp. 154/5/7 and 162) and Goldberg.
(pp. 149, 193 und 198).
[12] :
Cp. also the 365 of Enoch in Gen.5 .
[13] :
Enoch 72  80 ('The Astronomical Book'); Jubilees 6,32.36.38.
August 2008: And here it is.
To be continued...
With best wishes
Rüdiger Heinzerling