The numbers in the Book of Numbers, especially those of the so-called 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 twenty-three thousand of them died.", apparently hinting to Numbers 25,1-9 . 

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. 

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.3,40-51 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 non-zero 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.

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 

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.

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. 

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?

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.)

By the last transformation the sums became dividable by 365. Together they form the difference 
170 * 365 - 130 * 365 = 40 * 365  and can easily be interpreted in a number symbolic way.Being a difference of two (transformed) number columns, that are related to the beginning and the end of the desert journey by the Book of Numbers, the product 40 * 365 can be interpreted without constraint as the 40 years of the wilderness journey [10]. 

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.8.14.17-19). 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).

The striking frequency of numbers with equal decimals suggests that in constructing the 

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. - 

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.

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.

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 

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.

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.

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 

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.

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.