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)

Overview on numerical features in different scriptures

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