Reading the Torah with Equal Intervals

               a review by Prof. Daniel Michelson
                   Department of Mathematics
              University of California, Los Angeles




What is equal interval reading?

     Let us eliminate the spaces between the words  and  consider  the
text as a sequence of letters. Now, starting from a certain letter let
us skip N-1 letters and read the N-th one, again skip N-1 letters  and
read  the  N-th  one and so on. This will be called a reading with the
interval N. The number N may also be negative in which case the  read-
ing  is  backwards.  Of course, besides the interval N one has to know
where to start counting and how many intervals  to  go.  Let  us  make
things  more  clear  by showing a few examples. If one starts with the
first letter T in the Genesis (i.e. the T  of  BRA$'T)  and  skips  49
letters  one  arrives  at the letter ! in TH!M, again skips 49 letters
and arrives at R in !'RAH, again skips 49 letters and arrives at H  in
ALH'M.  Thus  one finds that the word T!RH (Torah) is spelled out with
interval 50 right in the beginning  of  Genesis.  The  number  50  has
several  important meanings in Judaism. The 50-th is the Jubilee year,
there are 49 days of Omer which are counted from  the  second  day  of
Passover  until the Shavuot which is the 50-th day, and there are also
50 gates of wisdom in Torah. The above example is a part of  a  bigger
pattern  found by Rabbi Michael Weismandel about 40 years ago (see at-
tached fig.1). Namely, in the second book Exodus,  the  word  T!RH  is
again  spelled  out with the interval 50 beginning with the very first
letter T in the book (i.e. the T of !ALH $M!T).  In  the  fourth  book
Numbers,  the  word  T!RH  is  spelled out with the interval -50, i.e.
backwards with the letter H starting in the first verse of  the  book.
Finally,  in the last book Deuteronomy, the word T!RH is spelled again
backwards however with interval -49 instead of -50 and  the  letter  H
starts  in the 5-th verse instead of the first. Why this deviation and
why is there no T!RH in the third  book  Leviticus?  Gaon  from  Vilna
wrote  in Aderet Eliahu that Deuteronomy actually starts from the 5-th
verse, while the first four verses correspond to the first four books.
Indeed,  the  fifth  verse reads: "On the other side of the Jordan, in
the land of Moab, Moses undertook to expound the Torah. He said ... ".
It  is claimed that Moses was given 49 gates of wisdom instead of fif-
ty. Since the subsequent explanation of Torah is given from the  mouth
of  Moses,  the word T!RH is spelled out with the interval -49. We see
that the system is symmetric - in the first two books T!RH is  spelled
out  forwards  and  in  the last two books - backwards.  Hence, in the
central book Leviticus you don't find the word T!RH. Instead, the four
letter  name  of G-d who gave the Torah is spelled out with interval 8
starting with the very first letter ' of Leviticus.( the number  8=7+1
is  closely  related to 50=7x7+1 but this is a separate story on which
we will not elaborate here).
     At this point a sceptical reader would  exclaim  that  the  whole
system  is nothing but a coincidence and the above explanation with 50
and 49 gates of wisdom was "cooked up" to tie  several  unrelated  ap-
pearances  of  the  word T!RH into a system.  "I'm sure", this sceptic
would continue, "you would be able to find such words and  systems  in
any book". Since the author of this review was, until recently, such a
sceptic -the question of coincidence versus intentional design will be
addressed  most  forcefully  in this article. Meanwhile let us mention
that on the statistical basis the word T!RH is expected to appear with
any  given  interval N in Genesis about 2 or 3 times. This estimate is
based on the total number of letters  in  Genesis(78064)  and  on  the
amount  of  the  letters  T(4152),  !(8448),R(4793) and H(6283) in the
book. Indeed, T!RH appears 3 times in Genesis  with  the  interval  50
which  is  what  one  would expect from any book of such length and of
similar concentration of letters T,!,R,H. There is however  no  reason
why  one of these three appearances should start with the very first T
of the book and why this should happen both in Genesis and Exodus.  As
a matter of fact the probability of such a coincidence is about 1 in 3
million!
     The above is one of hundreds of patterns found by Rabbi  Weisman-
del  in Torah in the time of WWII. After his death in 1948(?) his stu-
dents published in the early fifties the  book  "Torat  Chemed"  where
just  a  handful of his findings were exhibited. The rest of the find-
ings were lost. Of course at that time there were  no  computers.  In-
stead  Rabbi Weismandel was guided by a deep knowledge of Torah as for
what to seek and where to seek. As for the length of the  intervals  -
most of his examples refer to the numbers 50 or 26, the last being the
Gematria of the four letter name of God ('-H-!-H =10+5+6+5=26).  Later
on,  a  few  followers of Rabbi Weismandel continued the search, which
was still done by hand. We should mention Rabbi Shmuel Yaniv,  Abraham
Oren  and  their  students. But the real breakthrough occurred in 1982
when the computer was put to work. Here most of the credit  should  be
given to Dr. Eli Rips from Institute of Mathematics, Hebrew University
who was joined by Dr. Moshe Katz from the Technion, Haifa and later on
by  Doron  Viztum  from Jerusalem. Let us make it clear - the computer
does not have an intelligence to find meaningful patterns.  Instead it
is used as a fast and accurate counting machine. The text being inves-
tigated is typed into the computer and is stored there as  a  file  of
integers.  A  set of instructions would then tell the computer to look
for a certain word in the text with equal intervals in a given  range.
For  example, find all appearances of the word I$RAL- Israel (which in
integer form spells 10 21 20 1 12)  in  the  first  10000  letters  of
Genesis,  with  equal intervals ranging from -100 to 100. The computer
then shows that the word is spelled out only twice, the intervals  be-
ing  7 and -50 and is located in the four verses 1:31-2:3 (see fig.2).
We are stunned by the fact that these verses  constitute  our  Kiddush
recited  every Sabbath evening over a cup of wine. Indeed 7 and 50 are
the only numbers related to the Sabbath. The number 7 stands  for  the
seventh  day  of  creation and also for the seventh year - the year of
Shmita when the land rests. Then,  after  7  Shmita  cycles  the  land
should rest also on the Jubilee year - which is the 50-th year. Again,
a coincidence? A simple calculation shows that the probability of  the
word I$RAL to appear once with a given interval in the above verses is
about 1 in 1200. The chance of two appearances with intervals of 7 and
50  either  backwards  or  forwards is about 1 in 400,000. Another in-
teresting example is shown in fig.3. The text which is Gen.  38  tells
the  story  of  Yehuda and Tamar. As the result of their affair, Tamar
gave birth to Perez and Zerach. From the book of Ruth  we  learn  that
Perez started a lineage which led to Boaz. Boaz married Ruth and had a
son Oved, which had a son Yishai, which was the father of King  David.
So  it  was  a  natural  question  to  ask whether King David with his
lineage is hidden in this chapter. Indeed, you  find  the  names  BYZ,
R!T, YBD, '$' and D!D spelled out with the same interval -49, moreover
they all appear in the chronological order! We already  mentioned  the
importance of 49 being the 7-th Shmita followed by the Jubilee. Howev-
er 49 is also the last day of the counting of Omer which starts on the
second day of the Passover and ends a day before Shavuot. Every day in
this counting has a name and the 49-th is called MLK!T $BMLK!T - king-
dom of the kingdom. Is there a name which would fit David, the king of
kings, better? Let us also mention that David was born and died on the
very  day  of Shavuot and the book of Ruth is traditionally studied on
this holiday. But maybe this system is another coincidence? It is easy
to  estimate  the  probability of such an event. As we count the total
number of letters in Gen. 38 and the relative proportion  of  each  of
the letters of alphabet, we come to the conclusion that the probabili-
ty of the word BYZ to appear in our chapter with a given  interval  is
0.02.  (That is assuming that on the level of equal intervals the text
is random). Similarly, for the other four names the probabilities  are
0.63,0.065,0.76  and  0.25. The odds for all 5 names to show up with a
given interval are about 1 in 6,500. If we also request that the names
line  up  in  chronological  order,  the  chances  are reduced to 1 in
800,000. Now, if one would claim that the interval 49 is as  important
as  -49  and  the same for 50 and -50, these 3 possibilities would in-
crease the chances to 1 in 200,000 - still quite an impressive number!
     Let us turn to the third example in fig.4. We are in  the  begin-
ning  of  the  Parasha  !'CA  where it talks about the famous dream of
Jacob with the stairway reaching the sky.  As  Jacob  awoke  from  his
sleep  he  said,  "Surely the Lord is present in this place, and I did
not know it!" (Gen. 28:16). Where was this place? Rashi (the main com-
mentator  of  the  Torah)  writes that this was Mount Moriah where the
Temple was built later on. Moshe Katz who was reading the commentaries
decided to check for the word MQD$ - the Temple. Indeed, the word does
appear with a very important interval -26 starting with the M  of  the
word MQ!M (place) in the above verse. However as you continue to count
26 letter intervals after $ of MQD$ you find another five letter  word
HT!RH (the Torah) spelled forwards. Thus the two cornerstones of Juda-
ism HT!RH and MQD$ are spelled as one continuous sequence of 9 letters
with  the  interval  26  (which  is, to repeat, the numerical value of
Lord's name).  The probability of such an event (for a fixed  position
of  the first letter M) is about 1 in 17 billion! In the same story we
also find the words C'!N (Zion) and MQ!M (place) spelled out with  the
interval 26.
     The next example in fig.5 (found by Moshe  Katz)  is  related  to
Joseph's  second  dream (Gen. 37:9-10)- " Here I had another dream and
here the sun and the moon and eleven stars are bowing down to me".  On
which  Jacob  answers,  "What is the dream you have dreamed? Are we to
come, I and your mother and your brothers, and bow low to you  to  the
ground?".  Rachi explains what Jacob had on his mind: "the mother (the
moon) already died, while Jacob did not know that the moon  refers  to
Bilhah (Rachel's maid) who raised Joseph as if she was his mother". As
we stick together the words A$R XLMT HB!A (which you have dreamed...),
they  spell  RXL  MTH  ( Rachel died). Now we are looking for the word
BLHH (Bilhah).  The computer found two appearances of this word on the
same page, both starting with the same letter B next to the phrase A$R
XLMT - one is with the interval -99 and another with  -156.  We  don't
know exactly the meaning of 99 however 156 bears a direct reference to
Joseph being the Gematria of his name ('!SP = 10+6+60+80 = 156).
     There are hundreds of equally impressive examples which  are  not
shown  here  due  to the limited scope of this review. However, on the
basis of the presented material we ask again the same question  -  are
the above systems a mere coincidence or they are deliberately planned?
Now the sceptic concedes that the odds for each individual system  are
very  small, however there are millions of different stories which one
could look for so that occasionally some  of  them  occur  with  small
odds.  Likewise  in  a  lottery  there are millions of players and few
winners. The truth of the matter is that there are 3-4 people who have
been searching mainly the book of Genesis by computer for the last two
years. They explored perhaps a few thousand words  and  systems  while
the  success  ratio was astounding. Nevertheless, to counter the above
argument on a statistical basis one has  to  find  "story-independent"
phenomena, i.e. something which could be checked automatically by com-
puter and compared with other texts. The  following  example  will  be
used to demonstrate such a general phenomena. This example is also im-
portant from a historical perspective since it marked the beginning of
the "computer era" in the study of Torah.

A "hidden" Aaron in Leviticus

     Our story starts somewhere in 1982. Abraham Oren from kibbutz Sde
Eliahu was exploring manually whether the word AHRN (Aaron) is spelled
out with equal intervals in the beginning of Leviticus. Why Aaron  and
why  in  Leviticus?   As everybody knows, Leviticus talks mainly about
the work of the Cohanim - the priests, and Aaron being the Cohen Gadol
(the  high  priest)  is the main hero of the book. Nonetheless, in the
first open chapter (Parasha Ptucha) of Leviticus  Aaron  is  not  men-
tioned  even  once. Instead it repeats four times "the sons of Aaron".
Abraham Oren was familiar with the work of Rabbi Weismandel, so it was
natural  for him to suggest that Aaron is hidden inside the chapter in
the way of equal intervals. And indeed he found quite a few.  When  he
showed  it  to  Dr.  Eli Rips from Hebrew University, the latter typed
this chapter on the computer and asked it to find all  appearances  of
the four letter word AHRN in the chapter. The result of this search is
shown in fig.6. There are altogether 25 hidden  Aaron's  not  counting
the explicit ones.  The numbers which point to the circled A's are the
sizes of the intervals which should be counted from these A's in order
to obtain the word AHRN.  The negative numbers correspond to the back-
ward counting. In this example we are not selecting any  specific  in-
terval like 26 or 50. Instead the computer checks all intervals from 2
to 235 (the maximal possible in this chapter), forwards and  backwards
from every letter A and tries to find the word AHRN. As Rips looked at
the results he was overwhelmed by the large number  of  total  appear-
ances:  25. Indeed, the chapter is 716 letters long out of which there
are 55 A's, 91 H's, 55 R's and 47 N's. For a  random  distribution  of
these  letters a statistical formula shows that the expected number of
Aaron's in the text should be about 8  and  that  the  probability  of
finding  25  or more Aaron's  is about 1 in 400,000. That is, it would
take 400,000 pages of text like the one in fig.6 until one would  find
25  or more hidden Aaron's on a page. A linguist could charge that the
letters in the language are correlated so that the Hebrew of the Bible
may  "like" AHRN more then expected.  Notice, however, that 12 Aaron's
out of 25 are going backwards and it is not clear  why  the  "forward"
language should like them. And if it does, then equally well it should
like other combinations of the four letters A,H,R,N.  So Rips took all
12 possible combinations (there are 2x3x4=24, but forward and backward
count as one) and performed with them  the  same  experiment  as  with
Aaron.  In  the  lower part of fig.7 we see the results of the experi-
ment. The word AHNR (meaningless) appears in the text 8 times  and  so
does  ARHN. The other results 9,7,5 etc. center around 8 with a devia-
tion of +-3 in a complete agreement with the statistics and  only  the
AHRN  stands  out.  The  next experiment is shown in the upper part of
fig.7. As well known, in Hebrew there is a short and full spelling. In
Torah the same words sometimes are spelled full and other times short.
If we change the spelling, the equal intervals  become  at  once  non-
equal.  Hence  there  is no reason why the text should prefer  AHRN in
the form A(n)H(n)R(n)N over A(n)H(n+x)R(n+y)N. Now we fix the  numbers
x  and  y and let the computer to search for Aaron with all possible n
(i.e. from 2 to 235). The numbers x and y vary from -5 to  5  and  for
each  pair  x,y  the total number of Aaron's is shown in the table. We
see that these totals vary from 2 to 15 with the average 7.3  and  the
standard  deviation  2.4.  The  number 25 corresponding to x=y=0 (i.e.
equal intervals) is 7.4 standard deviations away from the average!  So
indeed,  our  text "likes" Aaron with equal intervals.  But what about
other words, maybe they exhibit the same phenomenon?  And  what  about
other  texts?  For  comparison Rips took all 4-letter words, more pre-
cisely all 4-letter combinations in Hebrew alphabet. Since  there  are
22  letters the total number of combinations is 22x22x22x22/2=117,128.
Now you take any word out of 117,128, say ABGD, and  do  with  it  the
same  experiment  as  with AHRN. Namely, you let the computer find the
number of times this word appears in  our  chapter  and  the  expected
number  of  appearances.   Suppose  that  for  ABGD  these numbers are
correspondingly 5 and 3. Then you compute the probability of having  5
or  more appearances of the word instead of the expected 3. The result
happens to be 0.185. Now turn to the upper table in fig.8. The  verti-
cal  axis shows the number of appearances of a word while the horizon-
tal - the probability (on a logarithmic scale). The number 232 in  the
6-th  row and 3-rd column shows that 232 words out of 117,128 appeared
5 times in the text and the probability for them was around 1/10,  and
similarly  for the other numbers. Thus the word ABGD was counted among
the 232. As the probability decreases and the  number  of  appearances
increases,  there are fewer and fewer words in the table. The position
of the word AHRN is shown by the circle. Obviously Aaron is the winner
of  the  competition!  There  is just one more word '+YA (meaningless)
with the same probability 1/500,000 which appeared 6  times.  Actually
all words with probability less then 1/1,000 turned out to be meaning-
less. There are also 12 words which appeared more times then AHRN, but
their  probabilities  are  quite reasonable. Indeed, there are more or
less frequent letters in  our  text.  The  words  with  very  frequent
letters  should  normally  appear  more  times.  But  what about other
texts? Rips took for comparison  a  piece  of  the  same  length  (716
letters)  from  the  beginning  of  the  novel "Hachnasat Kala" of the
famous Israeli writer Shai Agnon (the only Israeli Nobel prize winner)
and  ran on it the same experiment with 117,128 words. We see that the
distribution of numbers is the same as above with the only exception -
the  circle  which contained AHRN is now empty! And no meaningful word
passed the limit of the probability 1/1,000. This  proves  once  again
that  the  whole  phenomena  of AHRN has nothing to do with the Hebrew
language. But maybe the comparison with Agnon is unfair since his is a
different,  "modern"  Hebrew? Ideally, one should take a text which is
first - canonic, and second - very close to Torah. It was  Prof.  Ben-
Chaim from the Academy of Hebrew Language who came up with a brilliant
idea - take the Samaritan Torah! Samaritans are thought to be the des-
cendants of Kutim - the nations brought into Israel after the exile of
the 10 tribes ( 7-th century B.C.). Although they had been  influenced
by Judaism, they did not become a part of the Jewish nation. There are
still about 2,000 Samaritans living in  Nablus.  And  they  possess  a
Torah  which  differs from our tradition. Actually, there are numerous
differences among their manuscripts, so it is hard to  talk  about  an
established  version.  Nonetheless,  a  few years ago two Samaritans -
brothers Tzdaka, published the most authentic version of the Samaritan
text and compared it with our Torah. What is shown in fig.9 is the be-
ginning of Leviticus. The right hand side is our Torah,  the  left  is
the  Samaritan one.  Our chapter with Aaron's consists of the first 14
verses. The differences between two texts are  boldfaced.  Besides  an
additional  20-letter  phrase  in  the 10-th verse there are 16 places
where the texts disagree. But otherwise this is the same story and  in
translation  it would read the same. So it was very interesting to see
what effect these differences had on Aaron. And low and  behold,  they
destroyed  22  out of 25 hidden Aarons! However also 7 new Aarons sur-
faced. Thus the total became 10 instead of 25 - in complete  agreement
with  the  statistics since the expected number is about 8 with devia-
tion of plus or minus 3.
     At this point the sceptic is ready to  admit  that  people  could
have  done  it  deliberately.  "You know", he says, "they had a lot of
time to do this. The sages say that Rabbi Akiba used to count letters.
So apparently there was such a tradition".
     Let us explore this line of thought. Suppose some people, say the
priests  themselves  planted  these  Aaron's in the text. But for what
purpose? To impress somebody later on? However,  until  discovered  by
Abraham  Oren  and  Eli Rips this secret was absolutely unknown. More-
over, were it discovered 40 years ago nobody would be impressed by it.
Indeed  you  should  do all the comparisons to see how outstanding the
phenomenon is - and this was impossible before the advent  of  comput-
ers.  Did  the author(s) of the book anticipate the computer era?  And
then a technical question - how did they do it? Suppose there  was  an
existing text without Aaron's like the Samaritan Torah. Is it possible
with a little editing to create the 25 Aarons? The author of this  re-
view  actually  tried  to add another (26-th) Aaron to the existing 25
with no avail. But even if this is possible, there is a limit  of  how
many  words  one can hide in a meaningful text. The 25 4-letter Aarons
put 25x2=50 constraints on the 716  letter  text  (i.e.  the  distance
between  A and H is the same as between H and R and as between R and N
- giving two constraints per word). It is hard to set a precise  limit
but we feel that one can't produce a meaningful story where 30% of its
letters are tied up by constraints like those above. And this is not a
question of personal ingenuity or whether the author had a computer at
his disposal. The language has its set of words and grammatical rules,
so  mathematically speaking you are going to have more equations (con-
straints) then the unknowns (the words). Of course, if the  author  is
creating  the  language  simultaneously with the text - then the above
limit does not apply.
     These are indeed confusing questions. So our sceptic backs up and
suggests  that  maybe  the  whole  system with Aaron's is just another
coincidence. "After all, why did you take the first  chapter  and  why
Aaron?  There  are  so  many  chapters and so many important words you
could have chosen so that one success even with a ratio  of  1/400,000
is not outstanding at all!". We could reply that Aaron is the most im-
portant word in  Leviticus  and  intuitively  the  first  chapter  has
preference  over  the other ones. However the whole story with Aaron's
was brought here not for the sake of showing another oddity but rather
to demonstrate some general phenomena.

The clustering effect

     After the discovery of Aaron's, Rips obtained an electronic  text
of  Genesis  and  started a systematic investigation. (It was only re-
cently that the full electronic error-free text of Torah became avail-
able  to us). By the text of Torah, unless stated otherwise, we always
mean the traditional Ashkenazi Masoretic  text  as  published  in  so-
called  Koren  edition.  There is another text accepted among Yemenite
Jews. These two versions were carried by  two  independent  traditions
for  more  than a thousand years. Yet, as we compare these texts, they
differ only by 9 letters out of 304,805! Among the nine, there  are  3
different  letters  in Genesis (of a total 78,064). Besides, there are
several ancient manuscripts. One of them is called the Leningrad codex
(because  it  is  in  the  possession  of a Leningrad library) and was
copied 1,000 years ago in Egypt. As was shown  recently  by  Dr.  Mor-
dechai  Breuer  in "Keter Aram-Tzova" this text differs from the Koren
edition by 130 letters. Almost all of these 130 letters  are  contrad-
icted by the majority of other manuscripts and, most important, by the
Masoretic instructions. Nonetheless the Leningrad codex is called  the
"scientific  text"  of  Torah  and is used by several universities for
their databases. Clearly, even one missing or  extra  letter  destroys
the  hidden  words which "leap" over this letter. However the examples
shown in this review appear in parts of Genesis which  are  away  from
the doubtful letters and hence are not affected by them.
     So let us define the clustering effect. As we  saw  with  Aaron's
the  word  was  spelled explicitly (4 times) in the chapter and at the
same time it appeared there in a large concentration in the equal  in-
terval  form.  Rips wanted to check whether the same phenomenon occurs
with other words. Since it was not feasible to scan  all  words,  Rips
started with the words in the beginning of Genesis. The text in fig.10
consists of Gen.1 and 2 as it appears in the Koren edition. It  totals
2956  letters  and has about 120 different words of the length greater
than 2 (not counting different grammatical forms). Each word  was  run
by the computer to find where it appears with equal intervals. The in-
tervals n were taken in a range from 2 to some N,  both  positive  and
negative.  The  results  of  such a search for the word YDN (Eden) are
shown in fig.10. The word YDN  is  spelled  out  explicitly  in  three
places  as  shown by the rectangles. The circles show the hidden Edens
and the numbers leading to Y's indicate the appropriate intervals.  In
this  case  the range of intervals N was taken to be 120. The number N
is chosen in such a way that there is a reasonable  amount  of  hidden
words.  For  example, if one choses N=240 there would be twice as many
hidden Eden's mixing all over the text and it would  be  difficult  to
see  the  clustering.  Likewise  for N=60 it would be too few words to
make statistical estimates. We see that there are 4 hidden  Eden's  on
the  first  page  and 4 on the second page. The story of the Garden of
Eden is told in the verses Gen. 2:4-14 starting at the bottom  of  the
second page. Here inside a segment of 379 letters 16 hidden Eden's ap-
pear! What force has drawn them together? Maybe the 3 explicit  Eden's
increase  the local density of the letters Y,D and N so that there are
more chances for the hidden ones? A computation like the one performed
for  Aaron shows that the expected number of Eden's is about 5 and the
probability of such a deviation is about 1 in 10,000. (We see  another
weaker  cluster at the bottom of the third page where  the Torah tells
about the creation of the woman - indeed she was intended  to  be  the
YDN=pleasure for the man). In fig.11 we see a similar example with the
word HNHR - the river. The word is mentioned  4  times  explicitly  as
shown by the rectangle frames. When run on the computer with intervals
up to 80 it produces a cluster of 11 words over 2/3 of a page while on
a  usual  page it appears about 3 times. Next, in fig.12 the word MQ!H
(gathering of water) is exhibited. There is  a  cluster  of  10  words
around  the  explicit  MQ!H  while on the other pages the word appears
once or twice. Note that this time the hidden words do not  cross  the
explicit  one so that the letters of the explicit MQ!H could not cause
the cluster. Fig.13 demonstrates a similar effect with the  word  MQ!M
surrounded  by  a  cluster  of 8 hidden words, while on the second and
third page there are altogether 4 hidden words. Especially interesting
are  the  results for long words. Clearly, the longer the word is, the
smaller are the chances to find it in a text with a given interval. In
fig.14  three  such  words  are shown - BHBRAM (as they were created),
HX!'LH (Havilah) and HM!YD'M (the dates). The six-letter  word  BHBRAM
was  searched for by the computer over the whole book of Genesis (i.e.
78064 letters) with equal intervals in the range -300 to 300.  It  was
found  to  appear 4 times - one of them with the interval 176 clusters
around the explicit word. Similarly the word HX!'LH in the same  range
appeared  6  times - one of them with interval 167 clusters around the
explicit word. The seven-letter word HM!YD'M was searched for  in  the
book  of Genesis with intervals from -10000 to 10000! It appeared only
once, the interval being 70, and clusters  right  where  the  word  is
spelled  explicitly.  (By the way, there are exactly 70 days in a year
called M!YD'M as defined in Lev. 23 - 52 Sabbaths, 7 days of Pesach, 1
day  of Shavuot, 1 day of Rosh Hashana, 1 day of Yom Kippur, 7 days of
Sukkot and one day of Shmini Atzeret). But what about other words? Ob-
viously  we  cannot show here all of the results. However about 40% of
the words in the above 3 pages produced a  strong  clustering  effect,
another 40% showed a moderate clustering and the rest - no clustering.
Part of the clustering is effected by  the  non-even  distribution  of
letters.  For  example,  when the word ADM (Adam) is mentioned in Gen.
2:5,7 there is a nearby ADMH (the earth) which adds letters  A,D,M  to
the  text and increases the likelihood of the appearance of the hidden
ADM. When for comparison we took a 3000 letter piece of text from  the
novel  "Arie  Baal  Guf"  (The bulky Arie) of Bialik, there was also a
cluster effect although much weaker then in Genesis.  Hence, in  order
to measure the "net" clustering Rips suggested comparing the equal in-
tervals with the non-equal ones in the same text, as it was done  with
"Aaron"  (see fig. 7). The next question is - how to measure the clus-
tering quantitatively? The simplest way is to  specify  in  advance  a
neighborhood of the explicit word and then check how many hidden words
appeared in this neighborhood. It is clear, however, that  for  longer
words  the  neighborhoods  should be greater than for the shorter ones
and hence it is preferable not to compare words of different  lengths.
Finally,  a  controlled  experiment  was run for all 3-letter nouns in
Gen.1 and 2 - altogether 50 words. The neighborhoods to be  considered
were  300  letters  long  (about 8 lines) and centered at the explicit
words. The total number of hidden words in the neighborhoods  was  370
versus the expected 300, which was 4 standard deviations away from the
expectation. The results for non-equal intervals were about the  aver-
age. Next the same experiment was performed for the Samaritan version.
Here the results for the equal and non-equal intervals were about  the
same  as  the expectation.  Four standard deviations correspond to the
probability of about 1/100000. This is indeed  a  very  small  number.
However  some  statisticians may say that the text under investigation
is too short. Besides, for 3-letter words the non equal interval  test
is  very  limited. That is, for the word YDN we consider the sequences
Y(n)D(n+x)N with fixed x and all possible n. The number  x  should  be
small so that the non-equal intervals would be a small perturbation of
the equal ones. For example, if x varies between -5 and 5 we have only
10  different results to compare. If the word is longer, e.g. 5-letter
word ABCDE, the perturbed  sequences  are  A(n)B(n+x)C(n+y)D(n+z)E  so
that  with  x,y,z,  in  the same range of -5 to 5 there is a sample of
1330 different results. Hence Rips suggested to check  the  clustering
for  5-letter  words  over  the whole book of Genesis. This requires a
prohibitive amount of computations, so Rips restricted himself to  all
4-letter  nouns preceded by a definite article H which are encountered
in Genesis. The final list consisted of 86 words. Next  Rips  has  de-
fined  a  probability  function which measured the clustering for each
word. The definition is too technical to be  presented  here.  Roughly
speaking, the function attains  the values between 0 and 1, is uniform
for a random text and becomes small when a hidden word  with  a  short
interval  N  appears  close  to the explicit one. Then for each word a
"race" was performed in which the equal intervals  competed  with  the
non-equal  perturbations.  In  the first "race" the numbers x,y,z were
between -2 and 2 thus providing 5x5x5=125 "runners".  The  probability
function  was measured and the "runners" with the smallest value would
win. The results of the 86 "races" were as follows. In 3 instances the
equal intervals defeated the non-equal ones. The words were HMQNH (the
livestock), HXTMT (the seal) and  HBHMH  (the  domestic  animal).  For
eleven  more  words the equal intervals where among the top 10% of the
"runners".  These results are not impressive at all since  the  proba-
bility  that 14 out of 86 instances would be in the upper 10% is about
1/20. Next, the three winners were "allowed"  to  compete  with  about
5000 "runners". Namely, the range of x,y and z in the non-equal inter-
vals was increased from [-2,2] to [-8,8] which produced 17x17x17=4,913
"competitors". (It was too expensive to make such a "race" for all the
words since it takes several hours of computer time to  run  a  single
word).  The words HMQNH and HXTMT were champions also in the big race.
Now the combined phenomena of 14 top 10% words and 2 top .02% ones has
a probability of 1 over 30,000. The same experiment was performed also
with the Samaritan text. Here only two words - HQD$H (the harlot)  and
HMGDL  (the  tower)  were in the top 10% and no word entered the upper
1%. Thus the Samaritan text behaves like a "normal" one.
     Our sceptic might be unimpressed by the probability of  1/30,000.
Indeed,  with  Aaron's we already had 1/400,000. However this time the
test was both word and  segment  independent.  Namely,  instead  of  a
specific  (though important) word Aaron we took a big "natural" sample
and instead of the first chapter - the whole book of Genesis. One also
should  bear in mind that the clustering is only one aspect of the in-
finite information hidden in Torah in  the  way  of  equal  intervals.
There  is no clustering for "Torah" in fig.1 or for "Israel" in fig.2.
King David is not mentioned explicitly in fig.3  so  we  lose  another
story  and  likewise  for the "Temple" in fig.4 and "Bilhah" in fig.5.
One should really wonder that after all non-trivial patterns have been
neglected there is still something to observe.
     In the next section we  will  demonstrate  another  general  idea
which is common to many words and patterns.

The minimal intervals

     When the computer searches for a certain word with  equal  inter-
vals in a wide range of numbers it will find the word many times. Some
of the intervals may be of special interest  like  the  numbers  50,26
etc.  But  what  shall  we  do  with  the other ones? In the course of
numerous experiments Rips observed that the short intervals tend to be
more  significant  than  the long ones, i.e. they appear more often in
relevant places. We will present here one example of this  phenomenon.
The  text  in  fig.15 consists of Gen.2 (this is an enlargement of the
third page of fig.10). Verse 9 reads: "And from the ground Hashem  G-d
caused  to grow every tree that was pleasing to the sight and good for
food with the tree of life in the middle of the garden and the tree of
knowledge  of  good  and bad".  The names of the trees however are not
mentioned in the chapter. So Rips suggested that perhaps  these  names
are  hidden  in  equal intervals. The book of Yehuda Feliks "The fauna
and flora in the Torah" lists the names of all  the  trees  which  are
mentioned  in  Torah.  And  all  of these names  - a total of 26, were
found in the above chapter! Before the reader jumps out of  his  seat,
let  us explain that three- or four-letter words would normally appear
with some intervals in a segment as long as ours (about 1000 letters).
What  is  so  exceptional here - is that most of the intervals (except
for YRMN and LBNH) are very  short.  There  is  no  other  segment  in
Genesis  of  such  length  which contains so many trees with intervals
less than 20. Based on the density of the letters in the  chap
ter  one
could  estimate  the  probability  of  the  "orchard" phenomenon - the
number is about 1 in 100,000!

Conclusion

     We started with the "Torah" of Rabbi Weismandel, went through the
examples  of  "Israel",  "King  David",  "Temple-Torah",  "Rachel with
Bilhah" ,to "Aaron", then to the clustering effect in general  and  to
the  "orchard"  and  the  minimal intervals phenomenon. There are many
more fascinating examples and stories which could not be  included  in
this  limited review. A book with much of this material should soon be
published in Israel. We hope that our sceptic also concedes  that  the
equal  interval phenomenon is not an imagination of a few "phony" peo-
ple or a deliberate trickery with a computer but  a  reflection  of  a
hidden design. We are far from understanding the rules of this design,
in particular - what stands behind the numerical  values  of  all  the
different  intervals? In recent years there were some other coded sys-
tems discovered (or rediscovered) in the Torah.  Let  us  mention  the
multiples  of seven, where the key words in each chapter appear either
7 or 14 or 21 etc. times. Another rule discovered by  the  late  Rabbi
Suleiman  Sasson  states that for each word which is repeated in Torah
more than 80 times, its 80-th appearance is in a segment  which  talks
about a promise, covenant, marriage or purchase (i.e.  different types
of contract). The distinction of the equal intervals is that they  ap-
pear  on  the  letter rather then the word level and that they contain
apparently limitless information.
     But who made this design? Nachmanides writes in the  introduction
to  his  commentaries  on  Torah  that Moses saw the Torah as a letter
string of a black fire on the background of a white fire. This  string
of  letters  was  not divided into words. As G-d dictated the Torah to
Moses, he(Moses) wrote  it  accordingly  in  the  form  of  words  and
chapters.  As  Maimonides  states in the introduction to Mishne Torah,
Moses wrote the Torah before his death - one copy for each  tribe  and
one  to  be kept in the Ark. It is believed that the modern Torah text
is the exact copy of the original (modules maybe few letters, as  sug-
gested by the comparison of the Yemenite and Ashkenazi texts). This is
what Judaism claims.
     What do the Bible critics have to say? According to them Torah is
a  patchwork  which  consists  of pieces written in different times by
different authors. These pieces allegedly were put together during  or
after  the  Babylonian  exile and then canonized. For example they say
that Gen.1 and Gen.2 were written by different authors  because  Gen.1
uses  the name ALH'M=G-d while Gen. 2 the name Hashem G-d. Since there
are hidden words like BHBRAM in fig. 14 which connect Gen.1 and  Gen.2
we  should  assume  that  they  were built by the final editor. If one
counts all the trees in fig.15, the  most  outstanding  clusters  like
"Eden" and "the river" (fig.10 and 11) and few other systems with pro-
babilities less the 1/1000 - the number of  letters  employed  by  the
hidden  words  is about 30% of the total. Thus one has to believe that
this editor with some small modification  (and  without  any  apparent
reason)  created all these codes?  "It is possible", says our sceptic,
"that the ancients possessed some secret  knowledge  which  we  cannot
comprehend  -  take  for  example the great pyramids or the temples of
Inca". Whatever they knew, nobody would suggest that they could  fore-
see  the  future (unless they had a time machine?). We started with an
example of Rabbi Weismandel and we shall finish with  another  example
of his.  Everybody has heard the name Maimonides - the greatest Jewish
scholar and philosopher. In Hebrew his name is pronounced RMBM =  RaM-
BaM, the four letters being initials of Rabeinu Moshe Ben Maimon (Rab-
bi Moshe son of Maimon). Maimonides was born in Spain  851  years  ago
and  later  settled in Egypt where he became a court doctor of Tzalach
Ed-Din. There he wrote his most important  work  -  the  14  books  of
Mishne Torah where he classified and clarified all of the 613 Command-
ments - the 248 obligations and 365 prohibitions which are binding for
every  Jew.  Fig.16  shows  the  beginning  of  the Mishne Torah where
Maimonides explains what is the origin of  the  Commandments  and  how
they  are  divided  among his 14 books. There is a remarkable parallel
between Moses and Maimonides. They have the same name -  Moshe,  Moses
died at the same day he was born (Adar 7-th) and so Maimonides died at
the same he was born (Nissan 14-th). They both lived in Egypt and per-
formed  marvels before the rulers of Egypt (Maimonides as a court doc-
tor). Maimonides' Mishne Torah which is  a  full  summary  of  Judaism
parallels the Moses Mishne Torah, or Deuteronomy which is a summary of
Torah (see again the beginning of the review for  the  explanation  of
the "Torah" with interval 49 in Deuteronomy).  Furthermore, there is a
popular saying "M' M$H LM$H LA KM KM$H" - "from one Moses till another
Moses there was nobody like Moses". Nachmanides (RaMBaN) who lived few
decades after Maimonides claimed that he had  found  the  latter  once
mentioned in the Torah. The verse Exod. 11:9 reads : "Now the Lord had
said to Moses, Pharaoh will not heed you, in order that my marvels may
be  multiplied  in  the  land  of  Egypt" (see fig. 17). In Hebrew the
underlined phrase is RB!T M!FT' BARC MCR''M.  The  initials  of  these
four  words  form  the name RMBM = RaMBaM (which by itself consists of
the initials of the full name). "How  beautiful",  says  our  sceptic,
"but  you  probably  will find such RMBM on each page". We did check -
this is the single RMBM in the entire Torah spelled by the initials of
the  consecutive  words!  But this is only the beginning of the story.
Forty years ago Rabbi Weismandel came across this passage. And then he
asked  himself - could it be that there is something else about Rambam
hidden in a way of equal intervals? So he took the  name  of  Rambam's
greatest book Mishne Torah (spelled in Hebrew as M$NH T!RH) and start-
ed to search for it.  Since he already had discovered the "Torah" sys-
tem  with intervals of 50 (corresponding to the 50 gates of wisdom) he
was looking again for intervals of 50. And indeed, starting with M  of
M$H(Moses)  in  the  above mentioned verse he found the word M$NH with
the interval 50. The second part T!RH appeared much lower  again  with
interval  50.  The  large gap between M$NH and T!RH apparently puzzled
him.  He counted the number of letters between the M of M$NH and T  of
T!RH  - and it was 613 as the number of the Commandments. If one still
wishes to know the probabilities - the likelihood of  such  M$NH  T!RH
starting  with  a given M is 1 in 186,000,000. You could of course try
some other M, say 10 possibilities for M in a  close  neighborhood  of
the  RMBM.  And you could play with 613 counting them between the H of
M$NH and T of T!RH or M of M$NH and H of T!RH and also include or  ex-
clude  the first and the last letters in the counting, which gives you
6 possibilities. So with all this playing around you can increase  the
likelihood  to  1  in 3 million. Now, what is the bottom line?  Either
the one who  wrote  the  Torah  knew  2,500  years  in  advance  about
Maimonides  and Mishne Torah or the whole story is another coincidence
with a probability of 1/3,000,000. Unfortunately,  when  it  comes  to
very  small  or  large  numbers people often lose common sense. Let us
suggest a following mental experiment. One is offered  the  chance  to
play  Russian Roulette in which he loads the cylinder of a pistol with
one bullet out of 6 chambers, rotates the cylinder and shoots  at  his
head.  There  is  no  other  partner and one should repeat the game 81
times. If the person dies - he  dies.  If  he  stays  alive  (and  the
chances  are  1  in  3,000,000)  he  will have an exciting experience.
Would our sceptic take the offer?
     Three thousand three hundred years ago there was another  sceptic
-  Pharaoh  was his name. Our story in Ex. 11-12 is told after Pharaoh
had experienced nine plagues. He was still not convinced  because,  as
Torah  says, "The Lord had stiffened the heart of Pharaoh". Should one
wait for the tenth plague?
-----

     Here are some explanations for the attached material.  Since
we don't have Hebrew letter printer the Hebrew letters in the ar-
ticle have been represented by similarly sounding English letters
or similarly looking characters. The correspondence is as follows

  English letters/characters           Hebrew letters

            A                               Alef
            B                               Beit
            G                               Gimmel
            D                               Dalet
            H                               Hey
            !                               Waw
            Z                               Zain
            X                               Chet
            +                               Tet
            '                               Yud
            K                               Kaf
            L                               Lamed
            M                               Mem
            N                               Nun
            S                               Samech
            Y                               Ain
            P                               Pey
            C                               Tzadi
            Q                               Kuf
            R                               Reish
            $                               Shin
            T                               Tav

Please treat with respect the sheets with the text of  Torah.  If
you  wish to dispose them, they should be buried by Chevre Kaddi-
sha. If you don't know how to do it, bring it to some  synagogue.
The article is expected to be published in the Journal "Be-or Ha-
Torah", in English.
     The research of the codes in Torah and  the  publication  of
the  material requires  substantial financial support. We believe
that the discovery of the codes will have a strong impact on  so-
ciety  and  will  illuminate the eyes of those who are not indif-
ferent to the truth. If you wish to support the research,  please
send  your contribution to the foundation "Forum for Cultural and
Educational Exchange", 1100 S. Carmelina Ave.   Los  Angeles,  CA
90049. The number of the foundation for tax deduction purposes is
23 7134525.