Report on new ELS tests of Torah
            ================================

     Dror Bar-Natan, Alec Gindis, Aryeh Levitan, Brendan McKay

			29 May 1997

   ==== SUMMARY ====

   We have performed two series of experiments similar to that
   published by Witztum, Rips, and Rosenberg.  One matches the
   appellations of famous rabbis against the names of the books
   they wrote.  The other matches their appellations against the 
   years of their birth or death.  

   In each case, the result was unambiguously negative.
   No indication of any extraordinary phenomenon was found.
   

   ==== PROTOCOLS ====

   The following experimental protocol was published on 17 Apr 1997.

    1. Statement of Purpose

    Our aim is to further test the hypotheses made by Witztum, Rips,
    and Rosenberg in [WRR].  Several new lists of word pairs will
    be tested against the Koren edition of Genesis by two methods:

    A. A program identical in behaviour to Mr Rosenberg's program
       ELS2.C, with a permutation test equivalent to [WRR] for the 
       statistics P1 and P2.

    B. The following method suggested by Persi Diaconis.  For each 
       pair of persons p,p', compute one distance t(p,p') by averaging 
       the defined values c(w,w') where w is in the first word-set 
       of p and w' is in the second word-set of p'.  If there are no
       such values defined, t(p,p') is undefined.  For a permutation 
       pi of the persons, define T(pi) to be the average over all
       p of the defined values t[p,pi(p)].  If there are no such 
       defined values, T(pi) is undefined.  The result will be the 
       rank position of T(id) amongst all defined T(pi) for a large 
       set of random permutations pi.

    2. Principles

    Our subjects are chosen to permit as little subjective choice 
    as possible in the data preparation.

    We will use the reference encyclopaedia [EH] as our primary source 
    of data, with the less authoritative work [M] as a secondary 
    source.  In all cases we will use the data in [EH] unless it is
    obviously wrong, in which case we will use [M] to resolve the 
    error.  For other decisions we will follow the precedents set 
    by [WRR] wherever possible.

    The data for each experiment will be made available for 
    challenge, and the experiments will be run again if any error 
    is demonstrated.


    3. Experiments E1.1 and E1.2

    Experiment E1.1 will use the list of 34 rabbis and their
    appellations exactly as in Table 1 of [WRR].  The second 
    word-set for each rabbi will be generated from the year of
    birth and the year of death, according to these rules:

      R1.  Years will be taken from [EH].  If there is an obvious
           error in [EH], we will use [M] to resolve it.  We will
           also use [M] to assist when [EH] gives a year in the
           Western calendar but insufficient additional information
           to determine which of the two possible years of the 
           Hebrew calendar is correct.  However, following the 
           precedent of [WRR], we will not use any year which is 
           indicated by [EH] as being uncertain.

      R2.  According to the precedent established in [WRR], the 
           numbers 15 and 16 appearing as years within a century 
           will be expressed in two ways.

      R3.  Subject to the rules above, the list will comprise
           those words of 5-8 letters (precedent of [WRR]) formed
           from the year in each of these ways:

           Let yyy be the year within the millennium, and let myyy 
           be the same with the millennium indicated.  The following
           eight forms were approved by the linguist Professor 
           Michael Sokolov of Bar-Ilan University:

             F1:     yyy    
             F2:     Byyy    ('in yyy')
             F3:     $NTyyy  ('the year yyy')
             F4:     B$NTyyy ('in the year yyy')
             F5-F8:  The same as F1-F4 with myyy in place of yyy.

    Experiment E1.2 will be the same except that it uses the list
    of 32 rabbis and their appellations given in Table 2 of [WRR]. 

    
    4. Experiments E2.1 and E2.2

    Experiment E2.1 will use the list of 34 rabbis and their
    appellations exactly as in Table 1 of [WRR].  The second 
    word-set for each rabbi will contain the titles of his most 
    notable written works.

      R4.  Our definition of 'most notable written work' will be
           that the work is mentioned in both [M] and [EH] in the
           entry for that rabbi.

      R5.  The exact title as given in [EH] will be used unless
           there is a very clear error.  In the latter case, [M]
           will be used to correct the error.

      R6.  Subject to the rules above, the list will comprise those 
           titles containing 5-8 letters (precedent of [WRR]).

    Experiment E2.2 will be the same except that it uses the list
    of 32 rabbis and their appellations given in Table 2 of [WRR].

   On April 21, the following addition to the protocols was made:

    As separate experiments, we will also apply the same tests to 
    each of the other four books of the Torah.

   On May 1, the following request was received from Professor E. Rips, 
   and accepted as an addition to the experiment:

    "I would like to suggest (in addition to the procedure R3) to 
     consider the forms {F1,F2,F5,F6} (i.e. without $NT) separately 
     and to consider the forms (F3,F4,F7,F8) (i.e. with $NT) separately."

   To preserve the a-priori nature of the experiment, no further
   requests for additions or changes were accepted.

   
   ==== COLLECTION OF THE DATA ====

   Collection of the data posed no special problems.  The following
   is a summary of all cases where some unusual action was required.

   1. Rabbi Benvenisti:  [EH] gives the birth year as 5363 and the 
      death year as 5333, which is impossible.  We corrected this 
      error from [Mar], which gives the death year as 5433.

   2. Rabbi Margalit: His date of death is given as 12 Tevet and the
      year as 1780 (Gregorian calendar).  However, there was no
      12 Tevet appearing in 1780.  We corrected this error from [Mar], 
      which gives the year of death as 5541 (so he died on 9 Jan 1781).

   The actual data collected is given at the end of this document.

   As noted in the protocols, we will rerun the computations if
   any errors are demonstrated in the data AS DEFINED BY THE
   PROTOCOLS.  Piece-meal correction of errors using outside sources 
   will not be accepted, because non-systematic investigation is 
   known to be a fertile source a-posteriori bias.


   ==== DISCUSSION OF THE METHOD ====

   We have previously expressed criticism of Experiment A on various
   mathematical grounds.  However, since it was the method used in
   [WRR] (other than minor changes), we included it in order to make
   the present experiment independent of that debate.

   Experiment B has been severely criticised by E. Rips on the
   grounds that it does not satisfactorily measure the phenomenon
   he believes to occur in Genesis.  Essentially, he is concerned
   that the exceptionally small distances which occur occasionally
   may be masked by averaging them with a larger number of ordinary
   distances.


   ==== THE RESULTS ====

   We computed ranks out of one million permutations by calculating 
   the statistics for each of five million random permutations and 
   dividing the rank by five.  "B" refers to the statistic T(pi)
   defined in Experiment B.
   
                  Books      All year    Without   With
                               forms       $NT      $NT
   Genesis
     Table 1
        P1       946597       343991     268265   518287
        P2       897962       288110     079486   683097
         B       804395       063461     036526   232923 
     Table 2
        P1       227835       417339     834959   105783
        P2       268628       201746     720029   041576
         B       713015       244322     442305   033625
   
   Exodus
     Table 1
        P1       520579       708113     525718   740976
        P2       264919       701410     496007   747468 
         B       212843       529410     753949   012059
     Table 2
        P1       732340       906005     685038   916417
        P2       666454       537493     581435   458208
         B       553204       697032     421129   383316
   
   Leviticus
     Table 1
        P1       288488       929194     845731   847117
        P2       689177       945211     742176   915300
         B       440922       792107     986974   551347
     Table 2
        P1       191194       856053     778712   750315
        P2       073466       640612     653329   527906
         B       494075       935923     815783   802847
   
   Numbers
     Table 1
        P1       761420       390130     394191   444578
        P2       412348       353797     351793   442018
         B       569661       822768     474874   485556
     Table 2
        P1       305941       085703     188226   145228
        P2       467604       467653     631375   335369
         B       422428       711498     796720   584605
   
   Deuteronomy
     Table 1
        P1       612340       488540     752630   221857
        P2       770759       543284     738072   297303
         B       627681       677182     912213   604377
     Table 2
        P1       437418       473301     427928   526740
        P2       334035       422429     338638   546324
         B       192979       111412     554526   054611

   It is seen that the lowest value is 1.2%, produced by Experiment
   B for the Book of Exodus.  Considering the number of experiments 
   performed, this value is not small.

   
   ==== FURTHER COMPUTATIONS ====

   It must be stressed that none of the additional computations
   described in this section represent a-priori experiments.
   We will consider three matters.

   1.  The boundary between the two lists is an artifact of the
       history of [WRR].  Therefore it makes sense to consider
       the effect of using both lists together.

   2.  The strongest result in [WRR] was obtained after the
       removal of appellations starting with the word "Rabbi".
       Therefore it makes sense to try that here also.

   3.  On May 15, E. Rips requested that we use only the years
       of death, not the years of birth and death together.
       We did not agree to that change, but in any case we will
       present the results of that experiment here also.

   The results from the original experiment are included to make 
   comparisons easier.  Each rank is given like mmmmmm/nnnnnn, 
   where mmmmmm includes the appellations starting with "Rabbi", 
   and nnnnnn does not.  In the case of the years, we give two pairs.  
   The upper pair is the rank for both birth and death years 
   together, and the lower is the rank for the death year alone.

   We will only present the results for the Book of Genesis.
   The results for years of death in the other books are even 
   less interesting.

                 Books         All year       Without         With
                                forms           $NT            $NT
     Table 1
        P1   946597/786917  343991/417444 268265/309097 518287/567437
                            107933/188022 040073/059591 548529/622865
        P2   897962/657465  288110/261657 079486/156629 683097/511968
                            025506/032244 004150/020425 488451/310752
         B   804395/558328  063461/046783 036526/026419 232923/127553
                            010521/013124 010639/017908 400176/307306

     Table 2
        P1   227835/100008  417339/414092 834959/778628 105783/149229
                            677190/786334 819212/799814 346106/557166
        P2   268628/194173  201746/274949 720029/722988 041576/075552
                            434804/679920 626708/715350 268936/495786 
         B   713015/220562  244322/105014 442305/404187 033625/019805
                            379753/319462 375565/410462 302149/145498

     Tables 1 and 2 together
        P1   823043/463562  337511/362989 601181/536583 208691/289420
                            301703/449190 322849/315978 402008/621510
        P2   753366/437302  179017/190047 321917/389296 214528/186159
                            100186/189912 093055/191680 344982/382331
         B   848098/383244  079685/019538 108264/060895 038674/011613
                            050206/026570 051700/049921 291890/116418

   Here again we see no reason to claim other than chance behaviour.
   Removing of years of birth sometimes improves the result and 
   sometimes worsens it.  Similarly for removing the names starting
   with "Rabbi".

   The smallest value 0.4% is not very small considering the large
   number of computations we have performed.  In fact, a close look
   shows just how weak it is.  There are 72 defined values c(w,w') 
   for which w and w' belong to the same rabbi.  If they were 
   independent random variables with uniform distribution on (0,1), 
   the expected values of the smallest two would be 0.0137 and 0.0274.  
   The actual smallest two values are larger: 0.0172 and 0.0320.  
   Hence, this example certainly does not support the hypothesis that 
   very small distances are unusually common.  This conclusion is even 
   more inescapable if we remember that the appellations in this list 
   tend to produce below-random c(w,w') values even for random words w'.  
   Looking at c(w,w') values where w and w' belong to different rabbis, 
   we find 22 values better than 0.0172, including 9 perfect scores of 
   1/125.  It is hard to reconcile these facts with the score of 0.4%, 
   but it seems to be due to the small number of smallish distances 
   (8 at most 0.05) being unevenly distributed: there are 3 for rabbi 
   #22 and 2 for rabbi #5.  Removing rabbi #22 alone is enough to raise 
   the P2-rank by a factor of more than 8.

   
   ==== SOME OBSERVATIONS ====

   We begin with an observation that serves as a warning for 
   future experiment design:

      Rank orders out of 1 million.

      Genesis:     004311
      Exodus:      004948
      Numbers:     004071

   These consistently low values come from the famous books of the
   rabbis in Table 1.  What they measure is the rank order of the
   number of defined c(w,w') values, with large values taken as
   better than small values.  This statistic depends only slightly
   on the exact text, but more on its length and letter frequencies.  
   Most of all, these results are due to a built-in correlation 
   within the list of appellations and books.  Probably what is
   occurring is merely that the rabbis with more books to their 
   credit have been written about more and hence have more 
   appellations on average.
    
   Lest it be suspected that these results represent the discovery
   of a new phenomenon, we hasten to add that the same thing happens
   with randomly permuted Genesis texts.  Out of 45 random permutations,
   the best three scores (ranks out of a million) were 236, 762, 775.
   In order for c(w,w') to be defined, it is usually enough that some
   ELS for each word exists, irrespective of how many ELSs exist or
   where they are placed.  This is a very crude measure, in which
   the Genesis text is not known to be special in any way.

   In the case of the years of death, the strongest correlation of 
   this form (98%) occurs just at the place were the P2-rank is least.  
   It might be a coincidence, but the mathematics of the P2 statistic 
   is far too complicated to permit a satisfactory analysis.

   - - - - 

   In experiments like this, it is essential to follow the rules
   exactly, as otherwise the results are rendered meaningless.
   It has been thoroughly established that even a small amount of
   freedom in constructing the data can be exploited to obtain a
   result much better or much worse than it should be.  A case in
   point occurs for The Ramhal (#28 in first list).  We have omitted
   his book DRKH$M because it does not appear in either [EH] or [Mar],
   thus failing our criterion.  However, it is one of his most 
   famous books.  Such apparent anomolies cannot be predicted in
   advance and cannot be corrected a-posteriori without introducing
   an undesirable subjectiveness.  (As a matter of interest, 
   neither DRKH$M nor the alternative spelling DRKYHWH make any
   significant difference.)

   - - - -

   It is instructive to examine one of the perfect scores obtained
   for the experiment on books.  For the Book of Exodus we find
   c(HLBW$,LBW$YM)=1/125.  The reason is the one suggested by the
   words themselves: there is an ELS for HLBW$YM, which includes
   both HLBW$ and LBW$YM as substrings.  The chance of this
   happening is obviously much greater than 1/125.
   

   ==== REFERENCES ====

    [WRR] D. Witztum, E. Rips, and Y. Rosenberg,  Equidistant
          Letter Sequences in the Book of Genesis, Statistical
          Sciences Vol 9 (1994) 429-438.
    [M]   M. Margaliot (ed.), Encyclopaedia of Great Men of Israel.
    [EH]  Encyclopaedia Hebraica.


   ==== APPENDIX - The Data ====

   We will give the data here using the Michigan-Clairmont
   transliteration scheme.  The Hebrew alphabet in this scheme
   is )BGDHWZX+YKLMNS(PCQR$T.  Postscript files containing the
   data in Hebrew can be fetched from directory
   http://cs.anu.edu.au/~bdm/ELS.  The file names are books1.ps,
   books2.ps, years1.ps, and years2.ps.

   With the data for years, if only a single year is given it is
   the year of death.  If two years are given, the first is the
   year of death and the second is the year of birth.

   ---------------------------------------------------------------
   Books for Table 1.

   #1 has 2+2 words
    RBY)BRHM HR)BD   )SWRM$HW B(LYHNP$
   #2 has 1+1 words
    RBY)BRHM   M($HNSYM
   #3 has 4+13 words
    RBY)BRHM )BN(ZR) BN(ZR) HR)B(
    SPRH(BWR SPRHMSPR $PHBRWRH SPRH$M SPRYHWH SPRH(WLM SPRH)XD 
        YSWDMWR) )GRTH$BT SPRHYSWD $PTYTR SPRDQDWQ SPRHCXWT
   #4 has 3+4 words
    RBY)LYHW HBXWR B(LHBXWR   HHRQBH SPRHBXWR +WB+(M MTWRGMN
   #5 has 2+1 words
    RBY)LYHW HG)WN   )YLM$WL$
   #6 has 2+0 words
    RBYGR$WN HGR$NY
   #7 has 4+0 words
    RBYDWD DWDGNZ DWDG)NZ CMXDWD
   #8 has 3+4 words
    RBYDWD DWDHLWY B(LH+Z   +WRYZHB MGNDWD ZHBMZWQQ DBRYDWD
   #9 has 4+3 words
    RBYXYYM BN(+R )BN(+R )WRHXYYM   XPCH$M XPCYHWH PRYT)R
   #10 has 1+0 words
    RBYYHWDH
   #11 has 1+1 words
    RBYYHWDH   SPRHKBWD
   #12 has 4+6 words
    RBYYHWDH RBYLYW) HMHRL MHRLMPRG
    GWR)RYH NCXY$R)L DRKXYYM B)RHGWLH )WRXD$ NRMCWH
   #13 has 3+2 words
    RBYYWNTN )YB$YC B(LHTMYM   Y(RTDB$ $M(WLM
   #14 has 2+0 words
    RBYYHW$( RBYH($YL
   #15 has 2+1 words
    RBYYHW$( B(LHSM(   BYTY$R)L
   #16 has 3+3 words
    RBYYW)L SYRQ$ B(LHBX   BYTXD$ M$YBNP$ $WTHBX
   #17 has 0+3 words
    +WB+(M TWRTH)$M PR$THXD$
   #18 has 2+0 words
    RBYYWNH RBNWYWNH
   #19 has 7+3 words
    RBYYWSP YWSPQRW YWSPQ)RW MHRYQRW MHRYQ)RW BYTYWSP HMXBR
    $LXN(RWK BYTYWSP KSPM$NH
   #20 has 1+0 words
    B(LHCLX
   #21 has 1+0 words
    PNYYHW$(
   #22 has 2+1 words
    RBYY(QB RBNWTM   SPRHY$R
   #23 has 3+1 words
    RBYYCXQ )LPSY RB)LPS   TLMWDQ+N
   #24 has 3+0 words
    RBYY$R)L B(L$M+WB HB($+
   #25 has 2+0 words
    RBYM)YR HMHRM
   #26 has 4+1 words
    RBYMRDKY MRDKYYPH HLBW$ B(LHLBW$ LBW$YM
   #27 has 2+4 words
    RBYM$H )YSRL$   DRKYM$H TWRTX+)T MXYRYYN $WTHRM)
   #28 has 3+0 words
    LWC+W LWC)+W HRMXL 
   #29 has 2+3 words
    RBYM$H HRMBM   SPRHMCWT YDXZQH M$NHTWRH
   #30 has 2+0 words
    RBYCBY XKMCBY
   #31 has 4+5 words
    RBY$BTY $BTYKHN $BTYHKHN B(LH$K
    $PTYKHN H)RWK TQPWKHN PW(LCDQ MGYLT(PH
   #32 has 1+2 words
    RBY$LMH   SDWRR$Y $WTR$Y
   #33 has 4+4 words
    RBY$LMH LWRY) MHR$L HMHR$L
    YM$L$LMH XKMT$LMH (+RT$LMH $WTMHR$L
   #34 has 3+0 words
    )YDL$ MHR$) HMHR$)

   ---------------------------------------------------------------
   Books for Table 2.

   #1 has 5+1 words
    RBY)BRHM HR)BY HRB)BD HR)BD H)$KWL   )$KWL
   #2 has 3+0 words
    RBY)BRHM YCXQY ZR()BRHM
   #3 has 2+0 words
    RBY)BRHM HML)K
   #4 has 3+0 words
    RBY)BRHM )BRHMSB( CRWRHMR
   #5 has 1+0 words
    RBY)HRN
   #6 has 2+1 words
    M($YH$M M($YYHWH   YWSPLQX
   #7 has 2+0 words
    RBYDWD )WPNHYM
   #8 has 2+0 words
    RBYDWD DWDHNGYD
   #9 has 2+2 words
    RBYDWD DWDNY+W   M+HDN KWZRY$NY
   #10 has 1+6 words
    RBYXYYM
    (CHXYYM MQR)YQD$ YWSPLQX Y$R$Y(QB $BWTY(QB XNN)LHYM
   #11 has 2+3 words
    RBYXYYM BNBN$T   DYN)DXYY B(YXYY XMR)WXYY
   #12 has 4+0 words
    RBYXYYM KPWSY B(LNS B(LHNS
   #13 has 4+2 words
    RBYXYYM XYYM$BTY MHRX$ HMHRX$
    $WTSHRX$ TWRTXYYM
   #14 has 1+1 words
    XWTY)YR   XW+H$NY
   #15 has 1+0 words
    RBYYHWDH
   #16 has 2+5 words
    RBYYHWDH MHRY(Y)$
    LXMYHWDH BYTYHWDH BNYYHWDH M+HYHWDH $B+YHWDH
   #17 has 1+0 words
    RBYYHWSP
   #18 has 2+2 words
    RBYYHW$( MGNY$LMH   MGNY$LMH PNYYHW$(
   #19 has 9+2 words
    RBYYWSP M+RNY YWSP+RNY +R)NY M+R)NY MHRYM+ HMHRYM+ MHRY+ HMHRY+
    CPNTP(NX $WTMHRY+
   #20 has 3+3 words
    RBYYWSP T)WMYM PRYMGDYM   PWRTYWSP GNTWRDYM R)$YWSP
   #21 has 4+0 words
    RBYY(QB Y(QBBYRB MHRYBYRB HRYBR
   #22 has 2+3 words
    X)GYZ B(LHLQ+   (CHXYYM TXLTXKMH PTYLTKLT
   #23 has 8+1 words
    RBYY(QB MWLYN Y(QBSGL Y(QBHLWY MHRYSGL MHRYHLWY MHRYL HMHRYL
    $WTMHRYL
   #24 has 5+4 words
    HY(BC HRY(BC (MDYN HRY(MDN HRY(MDYN
    $)LTY(BC LXM$MYM MRWQCY(H MGYLTSPR
   #25 has 3+0 words
    RBYYCXQ HWRWWYC YCXQHLWY
   #26 has 4+0 words
    RBYMNXM QRWKML RBYM(NDL CMXCDQ
   #27 has 11+2 words
    RBYM$H ZKWT) ZKWTW M$HZKWT M$HZKWT) M$HZKWTW MHRMZKWT MHRMZ 
        HMHRMZ HMZLN QWLHRMZ
    $WTHRMZ TPTH(RWK
   #28 has 3+0 words
    RBYM$H MRGLYT PNYM$H
   #29 has 1+0 words
    RBY(ZRYH
   #30 has 2+2 words
    )XH(R Y$RLBB   M($HXW$B HWN($YR
   #31 has 6+2 words
    RBY$LWM MZRXY $R(BY $R$LWM MHR$$ HMHR$$
    )MTW$LWM NHR$LWM
   #32 has 1+1 words
    RBY$LMH   LB$LMH

   ---------------------------------------------------------------
   Years for Table 1.

   #1 has 2+10 words
    RBY)BRHM HR)BD
    TTQN+ BTTQN+ $NTTTQN+ DTTQN+ BDTTQN+ $NTTTP B$NTTTP BDTTP
        $NTDTTP B$NTDTTP
   #2 has 1+11 words
    RBY)BRHM
    BTTCX $NTTTCX B$NTTTCX DTTCX BDTTCX $NTDTTCX TTQMW BTTQMW 
        $NTTTQMW DTTQMW BDTTQMW
   #3 has 4+5 words
    RBY)BRHM )BN(ZR) BN(ZR) HR)B(
    TTQKD BTTQKD $NTTTQKD DTTQKD BDTTQKD
   #4 has 3+4 words
    RBY)LYHW HBXWR B(LHBXWR
    $NT$X B$NT$X $NTH$X B$NTH$X
   #5 has 2+10 words
    RBY)LYHW HG)WN
    BTQNX $NTTQNX B$NTTQNX HTQNX BHTQNX $NTHTQNX $NTTP B$NTTP 
        $NTHTP B$NTHTP
   #6 has 2+5 words
    RBYGR$WN HGR$NY
    $NTTNG B$NTTNG BHTNG $NTHTNG B$NTHTNG
   #7 has 4+9 words
    RBYDWD DWDGNZ DWDG)NZ CMXDWD
    $NT$(G B$NT$(G BH$(G $NTH$(G B$NTH$(G $NT$) B$NT$) $NTH$) B$NTH$)
   #8 has 3+10 words
    RBYDWD DWDHLWY B(LH+Z
    $NTTKZ B$NTTKZ BHTKZ $NTHTKZ B$NTHTKZ $NT$MW B$NT$MW BH$MW 
        $NTH$MW B$NTH$MW
   #9 has 4+10 words
    RBYXYYM BN(+R )BN(+R )WRHXYYM
    $NTTQG B$NTTQG BHTQG $NTHTQG B$NTHTQG $NTTNW B$NTTNW BHTNW 
        $NTHTNW B$NTHTNW
   #10 has 1+7 words
    RBYYHWDH   $NTQ+ B$NTQ+ $NTHQ+ B$NTHQ+ B$NTL $NTHL B$NTHL
   #11 has 1+5 words
    RBYYHWDH   TTQ(Z BTTQ(Z $NTTTQ(Z DTTQ(Z BDTTQ(Z
   #12 has 4+5 words
    RBYYHWDH RBYLYW) HMHRL MHRLMPRG
    $NT$S+ B$NT$S+ BH$S+ $NTH$S+ B$NTH$S+
   #13 has 3+6 words
    RBYYWNTN )YB$YC B(LHTMYM
    BTQKD $NTTQKD B$NTTQKD HTQKD BHTQKD $NTHTQKD
   #14 has 2+5 words
    RBYYHW$( RBYH($YL   $NTTKD B$NTTKD BHTKD $NTHTKD B$NTHTKD
   #15 has 2+5 words
    RBYYHW$( B(LHSM(   $NT$(D B$NT$(D BH$(D $NTH$(D B$NTH$(D
   #16 has 3+3 words
    RBYYW)L SYRQ$ B(LHBX   B$NTT $NTHT B$NTHT
   #17 has 0+10 words
    $NTTYD B$NTTYD BHTYD $NTHTYD B$NTHTYD $NT$L+ B$NT$L+ BH$L+ 
        $NTH$L+ B$NTH$L+
   #18 has 2+4 words
    RBYYWNH RBNWYWNH   $NTKD B$NTKD $NTHKD B$NTHKD
   #19 has 7+10 words
    RBYYWSP YWSPQRW YWSPQ)RW MHRYQRW MHRYQ)RW BYTYWSP HMXBR
    $NT$LH B$NT$LH BH$LH $NTH$LH B$NTH$LH $NTRMX B$NTRMX BHRMX 
        $NTHRMX B$NTHRMX
   #20 has 1+11 words
    B(LHCLX
    BTQNG $NTTQNG B$NTTQNG HTQNG BHTQNG $NTHTQNG $NTT(D B$NTT(D 
        BHT(D $NTHT(D B$NTHT(D
   #21 has 1+17 words
    PNYYHW$(
    BTQ+Z $NTTQ+Z B$NTTQ+Z HTQ+Z BHTQ+Z $NTHTQ+Z BTQYW $NTTQYW 
        B$NTTQYW HTQYW BHTQYW $NTHTQYW $NTTM) B$NTTM) BHTM) 
        $NTHTM) B$NTHTM)
   #22 has 2+5 words
    RBYY(QB RBNWTM   TTQL) BTTQL) $NTTTQL) DTTQL) BDTTQL)
   #23 has 3+12 words
    RBYYCXQ )LPSY RB)LPS
    BTTSG $NTTTSG B$NTTTSG DTTSG BDTTSG $NTDTTSG BT$(G $NTT$(G 
        B$NTT$(G DT$(G BDT$(G $NTDT$(G
   #24 has 3+5 words
    RBYY$R)L B(L$M+WB HB($+
    $NTTQK B$NTTQK BHTQK $NTHTQK B$NTHTQK
   #25 has 2+4 words
    RBYM)YR HMHRM   $NTNG B$NTNG $NTHNG B$NTHNG
   #26 has 4+5 words
    RBYMRDKY MRDKYYPH HLBW$ B(LHLBW$
    $NT$(B B$NT$(B BH$(B $NTH$(B B$NTH$(B
   #27 has 2+5 words
    RBYM$H )YSRL$   $NT$LB B$NT$LB BH$LB $NTH$LB B$NTH$LB
   #28 has 3+10 words
    LWC+W LWC)+W HRMXL
    $NTTQZ B$NTTQZ BHTQZ $NTHTQZ B$NTHTQZ $NTTSZ B$NTTSZ BHTSZ 
        $NTHTSZ B$NTHTSZ
   #29 has 2+11 words
    RBYM$H HRMBM
    TTQSH BTTQSH $NTTTQSH DTTQSH BDTTQSH BTTCX $NTTTCX B$NTTTCX 
        DTTCX BDTTCX $NTDTTCX
   #30 has 2+9 words
    RBYCBY XKMCBY
    $NTT(X B$NTT(X BHT(X $NTHT(X B$NTHT(X $NTTK B$NTTK $NTHTK B$NTHTK
   #31 has 4+10 words
    RBY$BTY $BTYKHN $BTYHKHN B(LH$K
    $NTTKB B$NTTKB BHTKB $NTHTKB B$NTHTKB $NT$PB B$NT$PB BH$PB 
        $NTH$PB B$NTH$PB
   #32 has 1+6 words
    RBY$LMH   BTTSH $NTTTSH B$NTTTSH DTTSH BDTTSH $NTDTTSH
   #33 has 4+5 words
    RBY$LMH LWRY) MHR$L HMHR$L
    $NT$LH B$NT$LH BH$LH $NTH$LH B$NTH$LH
   #34 has 3+15 words
    )YDL$ MHR$) HMHR$)
    $NT$CB B$NT$CB BH$CB $NTH$CB B$NTH$CB $NT$+W B$NT$+W BH$+W 
        $NTH$+W B$NTH$+W $NT$YH B$NT$YH BH$YH $NTH$YH B$NTH$YH

   ---------------------------------------------------------------
   Years for Table 2.

   #1 has 5+10 words
    RBY)BRHM HR)BY HRB)BD HR)BD H)$KWL
    TTQL+ BTTQL+ $NTTTQL+ DTTQL+ BDTTQL+ $NTTT( B$NTTT( BDTT( 
        $NTDTT( B$NTDTT(
   #2 has 3+10 words
    RBY)BRHM YCXQY ZR()BRHM
    $NTTP+ B$NTTP+ BHTP+ $NTHTP+ B$NTHTP+ $NTTK) B$NTTK) BHTK) 
        $NTHTK) B$NTHTK)
   #3 has 2+11 words
    RBY)BRHM HML)K
    BTQLD $NTTQLD B$NTTQLD HTQLD BHTQLD $NTHTQLD $NTTQ) B$NTTQ) 
        BHTQ) $NTHTQ) B$NTHTQ)
   #4 has 3+0 words
    RBY)BRHM )BRHMSB( CRWRHMR
   #5 has 1+11 words
    RBY)HRN
    BTQLB $NTTQLB B$NTTQLB HTQLB BHTQLB $NTHTQLB $NTTCW B$NTTCW 
        BHTCW $NTHTCW B$NTHTCW
   #6 has 2+10 words
    M($YH$M M($YYHWH
    $NT$MW B$NT$MW BH$MW $NTH$MW B$NTH$MW $NTR(G B$NTR(G BHR(G 
        $NTHR(G B$NTHR(G
   #7 has 2+10 words
    RBYDWD )WPNHYM
    $NTTCZ B$NTTCZ BHTCZ $NTHTCZ B$NTHTCZ $NTTKD B$NTTKD BHTKD 
        $NTHTKD B$NTHTKD
   #8 has 2+0 words
    RBYDWD DWDHNGYD
   #9 has 2+5 words
    RBYDWD DWDNY+W   $NTTPX B$NTTPX BHTPX $NTHTPX B$NTHTPX
   #10 has 1+9 words
    RBYXYYM
    $NTTQD B$NTTQD BHTQD $NTHTQD B$NTHTQD $NTTK B$NTTK $NTHTK B$NTHTK
   #11 has 2+10 words
    RBYXYYM BNBN$T
    $NTTLG B$NTTLG BHTLG $NTHTLG B$NTHTLG $NT$SG B$NT$SG BH$SG 
        $NTH$SG B$NTH$SG
   #12 has 4+0 words
    RBYXYYM KPWSY B(LNS B(LHNS
   #13 has 4+4 words
    RBYXYYM XYYM$BTY MHRX$ HMHRX$   $NTTZ B$NTTZ $NTHTZ B$NTHTZ
   #14 has 1+10 words
    XWTY)YR
    $NTTSG B$NTTSG BHTSG $NTHTSG B$NTHTSG $NT$CX B$NT$CX BH$CX 
        $NTH$CX B$NTH$CX
   #15 has 1+6 words
    RBYYHWDH   BTQL) $NTTQL) B$NTTQL) HTQL) BHTQL) $NTHTQL)
   #16 has 2+6 words
    RBYYHWDH MHRY(Y)$
    BTQK) $NTTQK) B$NTTQK) HTQK) BHTQK) $NTHTQK)
   #17 has 1+12 words
    RBYYHWSP
    BTTKZ $NTTTKZ B$NTTTKZ DTTKZ BDTTKZ $NTDTTKZ BT$CW $NTT$CW 
        B$NTT$CW DT$CW BDT$CW $NTDT$CW
   #18 has 2+4 words
    RBYYHW$( MGNY$LMH   $NTTX B$NTTX $NTHTX B$NTHTX
   #19 has 9+10 words
    RBYYWSP M+RNY YWSP+RNY +R)NY M+R)NY MHRYM+ HMHRYM+ MHRY+ HMHRY+
    $NT$C+ B$NT$C+ BH$C+ $NTH$C+ B$NTH$C+ $NT$K+ B$NT$K+ BH$K+ 
        $NTH$K+ B$NTH$K+
   #20 has 3+11 words
    RBYYWSP T)WMYM PRYMGDYM
    BTQNB $NTTQNB B$NTTQNB HTQNB BHTQNB $NTHTQNB $NTTPZ B$NTTPZ 
        BHTPZ $NTHTPZ B$NTHTPZ
   #21 has 4+4 words
    RBYY(QB Y(QBBYRB MHRYBYRB HRYBR   $NT$) B$NT$) $NTH$) B$NTH$)
   #22 has 2+9 words
    X)GYZ B(LHLQ+
    $NTTLD B$NTTLD BHTLD $NTHTLD B$NTHTLD $NT$P B$NT$P $NTH$P B$NTH$P
   #23 has 8+5 words
    RBYY(QB MWLYN Y(QBSGL Y(QBHLWY MHRYSGL MHRYHLWY MHRYL HMHRYL
    $NTQPZ B$NTQPZ BHQPZ $NTHQPZ B$NTHQPZ
   #24 has 5+6 words
    HY(BC HRY(BC (MDYN HRY(MDN HRY(MDYN
    BTQLW $NTTQLW B$NTTQLW HTQLW BHTQLW $NTHTQLW
   #25 has 3+6 words
    RBYYCXQ HWRWWYC YCXQHLWY
    BTQKZ $NTTQKZ B$NTTQKZ HTQKZ BHTQKZ $NTHTQKZ
   #26 has 4+0 words
    RBYMNXM QRWKML RBYM(NDL CMXCDQ
   #27 has 11+5 words
    RBYM$H ZKWT) ZKWTW M$HZKWT M$HZKWT) M$HZKWTW MHRMZKWT MHRMZ 
        HMHRMZ HMZLN QWLHRMZ
    $NTTNX B$NTTNX BHTNX $NTHTNX B$NTHTNX
   #28 has 3+6 words
    RBYM$H MRGLYT PNYM$H   BTQM) $NTTQM) B$NTTQM) HTQM) BHTQM) $NTHTQM)
   #29 has 1+9 words
    RBY(ZRYH
    $NTTZ B$NTTZ $NTHTZ B$NTHTZ $NT$L+ B$NT$L+ BH$L+ $NTH$L+ B$NTH$L+
   #30 has 2+10 words
    )XH(R Y$RLBB
    $NTTQG B$NTTQG BHTQG $NTHTQG B$NTHTQG $NTTMX B$NTTMX BHTMX 
        $NTHTMX B$NTHTMX
   #31 has 6+6 words
    RBY$LWM MZRXY $R(BY $R$LWM MHR$$ HMHR$$
    BTQLZ $NTTQLZ B$NTTQLZ HTQLZ BHTQLZ $NTHTQLZ
   #32 has 1+6 words
    RBY$LMH
    BTQM) $NTTQM) B$NTTQM) HTQM) BHTQM) $NTHTQM)


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