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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