Gray Code in the Solution of the Nine Rings Chinese Game

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1 Jourl o Mthemtcs Reserch; Vol. 9 No. 6; Decemer 7 ISSN E-ISSN Pulshed Cd Ceter o Scece d Educto Gr Code the Soluto o the Ne Rgs Chese Gme Erc Cesr Hug Shro Sherr Hug & Cheg-Hu Ts Morrso Acdem Tchug Tw Deprtmet o Egeerg Stord Uverst CA. USA Deprtmet o Mthemtcs Tchug Mucpl Tchug Frst Seor Hgh School Tchug Tw Correspodece: Erc Cesr Hug Morrso Acdem Tchug Tw. E-ml: cs77776@gml.com Receved: Septemer 7 Accepted: Septemer 9 7 Ole Pulshed: Octoer 7 do:.9/jmr.v96p URL: Astrct The m purpose o ths project s to eplore the e ch rged gme d to solve t through vrous ws cludg ducto d recursve methods. Assoctg ths gme wth the r codes d where the two umers represet whether the respectve rg s o the st row-the rg eg o the sword or the d row-the rg eg o the sword. Frst we eplored the prolem wth two mthemtcl models to d the estg ptters. The the usge o ducto we oud the geerl orm o the qucest umer o moves eeded depedg o the umer o rgs wthout the repetto o stuto. Hece we clled ths pth eutul soluto. Smlrl the usge o ducto we determed the smllest umer o steps eeded to get rom oe stuto to other stuto. Mewhle we lso ormulted orepetg sequeces to represet whch rg wll e moved t whch step o the eutul soluto s procedure. Fll we cocluded the project ggregtg the dt to geertg ucto. Kewords: e rgs Gr Code r code mtr recurso ucto sequece. Itroducto. Reserch Motves Ne rgs s cet Chese gme Wu ; Zhg W. & Rsmusse P.. Although the gme hs ee troduced or log tme umlr egers hve dcultes the opertos o the gme. To uloc t oe must ollow rgorous steps. I ths stud we oud ture o the r ucto Press et l 99 the gme s we eplored solutos steps whch we comed wth computer sotwre to crete ew d es operto model Hsu 969 ; Rosee J.A. & Rosee C.P.. We wll use the r to eplore the e rgs operto the coecto etwee e rgs d Gr code Grder 96; Kuo ; Wesste to crete ew method or e rgs operto model Mtrl-emples.com 9 d to eted t to three-stte gme.. Purpose o the Stud Frst we costruct mthemtcl model to help eplorg the e-rg soluto usg order rr grphs Schwrtz & Mthemtcs to eplore the somorphc reltoshp etwee the rr operto mthemtcl model d the e-rg soluto. Secod through the order rr grph mthemtcl model we vestgte the perect soluto o the e rgs d urther eplore the geerl solutos or rgs. Thrdl we dscuss the chrcterstcs o the trsormto ucto etwee the stte mtrces the order rr grph d the -t r umers d eplore the reltoshp etwee the rgs operto mode d ts stte chge. Fourth we eplore ts geertg ucto through the umer sequece recursve reltos ormed the umer o opertos the perect solutos or the rgs Youg Z. S..

2 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Reserch Methods. Costructo o the "rr operto" Mthemtcl Model.. Ne Rgs Soluto Stte Ne ch gme s to seprte the e rgs rom the m sword. Through phscl mpulto t s ot dcult to d tht ech ucle hs udmetl two sttes: whe the ucle hgg o the m sword d whe the ucle s releved rom the m sword. We reer t s the P stte pss stte d the F stte ree stte respectvel s show Fgure d : Fgure. All ucles re the P stte Fgure. All ucles re the F stte The so-clled "oe operto step" o the e rgs reers to ucle rom the P stte to F stte or rom the F stte to P stte oe mpulto. Ater the eperece lss o epermetl opertos we oud the ollowg pheome lsted s ollows: No mtter whether the ucle s the P stte or F stte ot ever ucle c chge ts stte. The stte o the outermost rst ucle c e chged regrdless o the ucle the P stte or the F stte. Thereore s log s the stte chge o the outermost rst ucle s completed we reer t s" T" operto Te operto. I the outermost rst ucle s the P stte the et ucle sde c e chged o mtter t s the P stte or the F stte. Oce the stte chge o the ucle s completed we reer t s "L" operto L operto. Complete the removl o e rgs rom the m sword requres serl operto process o the "T" opertos d "L" opertos... Estlsh Mthemtcl Model or Es Operto d -depth Dscusso Sce the operto o the e-rg props s ot coducve to prolem eplorto we tr to covert the props operto to the ollowg rr grph Fgure to smpl ech ucle operto d dee mthemtcl model ormto s ollows: Fg. Coverso o stte dgrms etwee e-rg d 9-order rr grphs

3 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Blc crcle the rst row o the rr grph represets the ucle o the e rg m sword the P stte. Blc crcle the secod row reers to the ucle o rom the m sword the F stte. The r sstem dctes the ucle stte o the e rgs props whch dctes ucle o the m sword d reers to ucles o rom the m sword. The postos o e ucles c e cotuousl represeted 9-t r umer. 7 smolzes e-ucle postos o the e-rg props soluto process. Smlrl the 9 mtr dctes the posto o lc crcle o the 9 order rr grph whch vector shows tht lc crcle s the rst row d reers to lc crcle the secod row. Thus the stte o the lc crcle o the 9 order rr grph c e descred s 9 mtr whch dctes the postos o e lc crcles o 9 order rr grph the operto process. Thereore t c e deduced tht there s oe-to-oe correspodece etwee the smolc represettos o these two sttes. Descrpto Through the correspodg gure trsormto ll the ucles o the e-rg m sword the 9 order rr grph correspod to lc crcle the rst row oe oe P stte. O cotrrl the ucles rom the e-rg m sword re mpped to the lc crcle the secod row oe oe the 9 order rr grph F stte. Thereore the stte o the e-rg ucle s cosstet wth the lc crcle stte o the 9 order rr grph. The correspodece etwee these two sttes s oe-to-oe. For ucle stte chges the e-rg props there re two opertg uctos "T" d "L" opertos. For emple the stte Fgure s chrcterzed -t r represetto.the T d L Becuse the stte o the ucle o the e-rg props s somorphc to the stte o the 9 order rr grph the ollowg correspodece s lso stsed: T ; L Thus there s oe-to-oe correspodece etwee the opertos the lc crcle stte chge o the 9 order rr grph d the stte chge soluto o the ucles o the e-rg props. "T" operto c e perormed o the rst d secod ucles to the rght Fgure where "L" operto c e doe o the ourth ucle. The sttes o the d th 6th 7th th d 9th ucles c t e chged. I cocluso we d tht the e-rg soluto s trsormed to the stte chge model o the 9 order rr grph t s equvlet to dscuss whether there s seres o "T" d "L" opertos tht chges the tl stte mtr to uloced stte mtr?.. Eplortos o the Opertol Propertes o Mtr Stte Chge Net we wll dscuss the ollowg ew propertes the "T" d "L" opertos o the lc crcle stte chge the ew mthemtcl model o 9 order mtr grph. 6

4 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Propert. Besdes these two stte mtrces d ll other stte mtrces c use T operto d L operto. s med the "rst etreme stte mtr" d the "secod Etreme stte mtr". Proo There re 9 lc crcle sttes o the 9 order rr grph show the gure. Sce the deto o L operto s to chge the lc crcle stte et to the rght outermost lc crcle the rst row regrdless whether ths lc crcle s locted o the rst or secod row. It s cler tht o the mtr smols the rst etreme stte mtr s ule to coduct L opertos ecuse there s o lc crcle o the rst row. Sce there s o lc crcle sde the rst lc crcle o the rst row o the secod etreme stte mtr t c t e operted wth L operto. The rest o the stte mtrces re cosstet wth the T d L opertos the stte chge deto so the c perorm T operto d L operto. Propert. T d L opertos re reversle the stte chge o the 9 rr grph. Prove Let X e lc crcle stte o the 9 rr grph X = the T T T Let X e lc crcle stte o the 9 rr grph wthout loss o geerlt X = L L X X. T T X X. the L L L Propert. The stte chge operto the perect soluto o the e rgs o the 9 rr s composed o T d L opertos ltertel. I order to cltte the susequet dscusso we me the ollowg detos: Deto I the 9 rr grph the X stte s chged to the Y stte ter the T operto or L operto TX=Y or LX=Y. It s deed s "segmet" or the X stte to the Y stte. 7

5 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 I the stte trsto o the 9 rr grph the X stte s chged to Y stte v seres o T or L opertos T L T L... T X = Y or T L L... L X = Y. We cll t "pth" or the X stte to the Y stte. A " segmet" s lso sgle "pth". The umer o T or L opertos tht c e mde to stte o oe pth the 9 rr stte chge process s clled the "degree" o ths stte. For emple the degree o the two etreme sttes s d other stte s dmesos re. Propert. There s sgle pth or chge etwee two sttes rom ll sttes o the 9 rr grph. Alss There re lc crcle sttes the 9 rr grph where the degree o the two etreme sttes s the degree o the remg sttes s d rom propert the stte s chged oth T d L ltertg operto. Accordg to the prcple o oe stroe rom the rst etreme stte s strtg pot the rst operto s T operto d ter seres o L d T ltertg opertos we c rech to the ed pot the secod etreme stte. The rst d lst opertos re T opertos. The totl umer o opertos s odd umer d the totl umer o ll sttes s eve umer. I theor ths pth should est d s uque. So we me the ollowg deto: Deto I there s pth rom the rst etreme stte ter seres o ltertg T d L opertos to the secod etreme stte. Ths pth coects ll the sttes o the order rr grph. We cll ths pth the "whole pth" o the order rr grph. I the precedg proo process some o the used smols re deed s ollowgs: Deto I ll stte mtrces o order rr grphs the stte mtr o ech colum vector s the sme s we cll thus the rst etreme stte mtr X. I ll stte mtrces o order rr grphs the stte mtr o ech colum vector s the sme s. We dee ths the ll upper stte mtr Z.

6 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 I the stte mtrces o the order rr grph ecept or the rst colum vector the rest o colum vectors re ll stte mtr whch s deed s the secod etreme stte mtr Y. I the stte mtrces o the order rr grph ecept the secod colum vector the rest o colum vectors re ll stte mtr we cll ths the rst echge stte mtr V. I ll stte mtrces o the order rr grph ecept the rst d secod colum vectors the rest o colum vectors re ll stte mtr we cll ths the secod echge stte mtr U From the ove lss proo o propert s equvlet to provg tht there s sgle d uque ull pth the stte chge operto o the order rr grph. Proo Usg mthemtcl ducto = there re two sttes d respectvel. Ater T operto ormg uque pth = holds. = there re our sttes d respectvel. Through two T opertos d oe L operto t orms uque pth whch s T T Assume tht P. L. So = holds. t holds. I the stte chge operto process o order rr grph there s ol ull pth X s strtg stte d Y s the ed stte. Ater tmes o ltertg T L T T opertos t orms the ol ull pth. I the stte chge operto o the + order rr grph let Stte X X 9

7 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Stte Stte Stte Y X U Y V Y d the stte X e strtg pot. Ater the stte chge operto o the ull pth P t c e coected to the stte U. The stte U c e coected to the stte V ter oe L operto. Ater the reverse operto o ll the stte chges t the ull pth P the stte V s coected to Y. There s totl tmes o ltertg opertos T L T T. I cocluso the uque ull pth the stte chge operto o the + order rr grph s X U V Y. So t lso holds. P L reversep Through the mthemtcl ducto or ll postve tegers there s sgle ull pth the stte chge operto process o order rr grph. Ths propert pls mportt role the ollowg qur process d there s lso rgorous rgumet process. We sort t to Theorem s ollows: Theorem. For ll postve tegers d ll the stte mtrces o the order rr grph the two etreme stte mtrces re used s strtg d ed pots ter seres o ltertg T d L opertos we c hve ollowg two results: I the stte chge operto process o order rr grph there must e sgle ull pth. The umer o opertos requred to complete ths ull pth s strtg wth T operto d edg wth T operto... Eplore the Geerl Reltoshp etwee the Stte Chges o the Full Pth o Arr Grph.. The stte mtr reltoshp etwee the rr grph d the + order rr grph: Let the stte mtrces the rr grph e A mtrces A re serted to the colum vector. Acordg to Theorem the stte o the let sde o the rst le we hve the ew stte mtr A.

8 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Isertg to the colum vector o the let sde o the rst le we hve ew stte mtr A whch orms the stte mtrces. These re the stte mtrces o the + rr grph.. Reltoshp Betwee the Full Pths o Arr Grph d + Arr Grph. We dee the whole pth o the su-stte chge opertos the order rr grph s P. Accordg to theorem ths whole pth s ectl the rst hl o the whole pth P d the pth cots the stte mtrces A o ll the rst colum vectors. The secod hl s ectl the reverse o the whole P d cots the stte mtrces A pth o ll the vectors the let. Besdes ter oe L operto the stte mtr the mddle prt o the whole pth the rst colum o P U = c e trsormed to V =. Thereore the operto process o the ull pth P the + order rr grph s s ollows: the stte mtr X s chged to Y ter the whole pth P operto the to Y ter oe L operto d ll to X v the whole pth P reverse operto.. Summr: From the dscusso ove we summrze the ollowg results: The whole pth stte chge operto c rrge stte mtrces o the order rr grph to sgle pth trsormto mp G. For emple the whole pth stte chge operto P c e rrged to sgle pth trsormto mp G : T L T L T L T

9 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 O the oe stroe mp G or the stte mtrces o the ull pth P msg the rst vector the rst hl o the stte mtrces s ectl the stte mtrces o oe stroe mp G or the ull pth P the sme order. It reches the stte mtr Y d through the L operto t ecomes Y. It the cotues the ltter hl o the stte mtrces. The order o ths ltter hl stte mtrces s the reverse order o the stte mtrces o the oe stroe mp G. Tht mes the re smmetrcl to the rst hl stte mtrces. Thereore the oe stroe mp G or the ull pth P s: X P Y L Y reversep X c I the stte chge operto process o order rr grph there must e shortest pth etwee two stte mtr A d B. Thereore the perect soluto o the e rgs Chese gme s the shortest pth the whole pth P. 9 d The process o reverse stte operto the perect soluto o the e rgs gme s equvlet to the oe stroe mp o the whole pth P 9 show s ollows: 9 Z prtlpthop L reversep Z We summrze t to Theorem. Y Y X Theorem For ll postve tegers there s uque ull pth P ll the stte mtrces o the order rr grph there must e the ollowg two results: A B Gve two stte mtrces d there must e shortest pth chgg the stte mtr to stte mtr B through ltertg T d L opertos. P The stte mtrces etwee the rst hl d the secod hl o the ull pth re smmetrcl. The stte mtrces o the rst hl re chged to those o the secod hl v oe L operto.. Eplore the Perect Soluto o the Ne Rgs Chese Gme d Geerlze the Perect Soluto o the Rgs Gme As we ow tht the e rgs perect soluto s to rele the e ucles rom the m sword wth the lest umer o opertos. Usg our prevous dscusso comed wth the stte mtr presetto the mthemtcl model o order rr grph we c see tht the process o solvg the perect soluto o e rgs s equvlet to chgg the A whole upper stte mtr to the rst etreme stte mtr through seres o T or L opertos. Ths operto perectl mtches the reverse stte chge operto the prtl prt o the ull pth P 9 o the order rr grph. So The umer o steps or the perect soluto o the e rgs gme wll e less th 9 -. We c use the "ull pth" model to eplore the e rgs soluto d eve the geerl soluto o the rgs.

10 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7.. Ne Rgs Perect Soluto d ts Numer o Steps Usg Theorem Theorem d verse deducto method we eplore the umer o steps to solve the e rgs: Frst let e the totl umer o T d L opertos eeded to chge the whole upper stte mtr to the rst etreme stte mtr the ull pth stte chge operto P o the order rr grph. Ovousl Z 9. Bsed o Theorem the perect soluto o the e rgs s to chge the ll upper stte mtr to the rst etreme stte mtr X 9 through 9 tmes o serl T d L opertos the reverse stte chge opertos o the ull pth P 9. Accordg to Theorem there must e shortest pth two stte mtrces o the ull pth P 9. Ater seres o T d L opertos the stte mtr Z 9 s coected to the stte mtr X 9. Smlrl ccordg to Theorem whle the stte mtr Z 9 s trsormed to the stte mtr X 9 the ltter hl process s ectl the ull pth P. The totl umer o tmes ltertg T d L opertos equls to 6. I ddto lzg the stte mtr represettos o the rst hl pth wth the rst d secod vectors gored we oud tht the rst hl o the pth s ectl the process or the stte mtr Z 7 trsormed to the stte mtr X 7 whch the totl umer o T d L opertos s 7. 6 From the ove ccomprehesve dscusso we hve d. 7 I ddto we c e erred tht 6 6 d. Sce d we hve the recursve reltoshp N Usg the ove recursve relto d the geometrc seres ormul we c deduce So t tes steps to completel solve the e rgs. The geometrc seres presetto epresses the perect soluto process o the e ch s ollows : T recursvep steps recursvep steps Z Z X Z X Z X 9 recursvep6 steps 6 6 recursvep steps Z X X 7 We rst do the L operto the the reverse operto o the whole pth P P P 6 P sequece rechg the l stte mtr X 9. The e rgs re completel uloced rom the m sword... N Rgs Perect Soluto d the Numer o Steps. Applg the prevous vestgto methods d results e rgs gme we the eplore the umer o rgs opertos steps: Frst the umer o steps the perect soluto o the rgs stses the recursve reltoshp: 9 N The ollowgs re the geerl ormuls or s odd: wth odd d eve umers respectvel:

11 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 d so [ ] From the ove recursve reltoshp d the geometrc seres ormul we c deduce tht the rgs perect soluto process wth odd umer s ollowgs: T recursvep steps recursvep steps Z Z X Z - X Z X - - recursvep - steps recursvep- steps Z X Z X X Smlrl whe s eve: d so [ ] From the ove recursve reltoshp d the geometrc seres ormul we c deduce the ch perect soluto process wth eve umer s ollowgs: L recursvep steps recursvep steps Z Z X Z X Z X recursvep - steps recursvep- steps Z X Z X X From the [ results o d we hve the geerl ormul ] Theorem. Theorem. 6 N d summrze t s For ll postve tegers the rgs perect soluto d the umer o soluto steps ollow three propertes. the umer o steps to complete the perect soluto meets the recursve reltoshp: N The geerl orm or s [ ] N. Whe s odd the perect soluto process s: T recursvep steps recursvep steps Z Z X Z X Z X -

12 Jourl o Mthemtcs Reserch Vol. 9 No. 6; recursvep - steps recursvep- steps Z X Z X X Whe s eve the perect solutos process s : L recursvep steps recursvep steps Z Z X Z X Z X recursvep - steps recursvep- steps Z X Z X X. Ivestgte the propertes the Trsormto Fucto etwee the Order Arr Grph Stte Mtr d the -t Br Numer. We wll eplore the shortest pth etwee two sttes o the e rgs gme d whch ucle s operted the mth step rom oe stte to other. Further we wll geerlze the operto process or emple chs whether we c uloc ll the ucles stte or wht stte t wll ecome ter movg m steps rom stte A?.. Fd the Trsormto Fucto etwee the Order Arr Mtrces d the -t Br Numers. Accordg to Theorem there must e sgle uque ull pth the stte chge operto o the 9 order rr grph d ths Euler pth just coects the 9 stte mtrces through seres o T d L opertos ltertel. We wt to use ths s ss comed wth the -t r represetto to eplore whether there s trsormto ucto mppg the stte mtr to correspodg -t r umer. Alss: Frst we hve two deult trgets or ths trsormto ucto. Oe s tht t s oe-to-oe mppg ucto. The other s tht t c relect ll stte mtrces postos the order rr grph to tht o the ull pth the sme order d the order o the postos s ectl the sme s the order o the correspodg ts umers. Net the reltoshp etwee the stte mtrces d the r umer trsto o the ull pth o the order rr grph s dscussed wth =. Whe = the stte mtrces o the order rr grphs re cocteted. I we ocus o the umers the rst row the stte mtrces correspod to d.e. oe dgt o r represetto. Whe = the stte mtrces the ull pth o the order rr grph re 6 T T L. I we ocus o the umers the rst row o mtrces the stte mtrces correspod to.e. -t r represetto. The posto order o the mtrces d the correspodg umer order r presetto re ot cosstet. To me them cosstet we must tret s d respectvel. Thereore we c ot ol ocus o the umers o the rst row. It s ecessr to reer to the etres the secod row. We rst tret the umer the rst row d the rst colum s the oe the rst colum d the secod row posto d the do the "XOR" operto usg ths ew vlue d the etr the secod colum d the secod row. Thus respectvel. correspods to the r d correspods to the r. These two vlues re d At ths tme the stte mtrces correspodg to.e. o the -t r umers d the order o ther postos d the order o ther correspodg r umers re cosstet.

13 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Whe = the stte mtrces o the order rr grphs s cocteted s T L T L T T L Mod the correspodg reltoshp or the trsormto we te the vlue the rst row o the rst colum or the upmost dgts d the use etres o the prevous row the rst colum d the oes o the two cosecutve rows ehd the rst colum or the "XOR operto. The stte mtrces respectvel correspodg to.e. 67 o the -t r umers. Ther posto order d the oe o the r umers re cosstet. Bsed o the ove dscusso trsormto ucto s oted whch ssoctes the stte mtr A wth 9-t r umer 7 6 whch XOR 7 Z. Tht s. A Bsed o the result o the ove lss d dscusso we descre d prove the Theorem. Theorem. I there s trsormto ucto etwee ll stte mtrces d -t r vlues o the ull pth o the order rr grph mps the stte mtrces A.Ths ucto A to the -t r umers where XOR. Z The stte mtr trsormto ucto s oe-to-oe mppg ucto d strctl cresg ucto. test From the ove results t s ow tht there s trsormto ucto etwee the set M ormed the stte mtrces coected the ull pth o the 9 order rr grph d the set N ormed the 9-t r vlues. Ths ucto s oe to oe d surjecto ucto. The stte mtr o the 9 order rr grph s somorphc to the 9-t r vlue. We c use ths ucto to chec the umer o steps requred or the perect soluto o the e ch gme: 6

14 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 The ull upper stte mtr or egg the e ch gme s Z 9 d the l stte mtr or solvg the e ch gme s X 9. The postos o these two mtrces the whole pth P9 re 6 Z d X 9 Z 9 X 9 9 respectvel. Sce the umer o steps to perectl solve the e ch s steps.. Propertes o the Trsormto Fucto etwee Order Arr Grph Stte Mtrces d -t Br Numers We urther eplore some propertes o the trsormto ucto such s the meg o the ucto vlue prt d the verse ucto. Prt o the ucto vlues. Bsed o the prevous stud results we ow tht the ucto etwee the 9-t r vlues d the correspodg stte mtrces o the ull pth o the 9 order rr grph s strctl cresg. Thereore studg the order the 9-t r vlues o two stte mtrces we c determe the seres o T d L opertos etwee these two mtrces. The ollowg s dscusso o the process: Sce there re stte mtrces o the ull pth o the 9 order rr grphs t s ecessr to use seres o - tme T d L lterte opertos whch the egg s the T operto d the edg s lso T operto. Thus these odd umer opertos re T opertos d the eve umer opertos re L opertos. I the ull pth o 9 order rr grph esdes the rst d th stte mtr the odd- umer stte mtr hs T operto orwrds d L operto cwrds. I the other hd the eve-umer stte mtr hs L operto orwrds d T operto cwrds. I ddto the stte mtr trsormto ucto wth oe-to-oe propert hs 9-t r vlues. Thereore stte mtrces o the ull pth o the 9 order rr grph re represeted the trsormto ucto s.... Thus the odd-umered stte mtr ucto vlue s eve whch does T operto orwrds d L operto cwrds. The eve- umer stte mtr ucto vlue s odd whch does L operto orwrds d T operto cwrds. We wll dscuss the results s Theorem d prove t. Theorem. Through the mtr trsormto ucto o the ull pth o the order rr grph rtrr stte mtr A s trsormed to -t r vlue A.Let d e rst d secod etreme stte mtrces o the ull pth o the order rr grph. The: Z Y I s the stte mtr o the ull pth P d A Z the vlue A s. The stte mtr A s coected to the secod etreme stte mtr v seres o the order o Y T L T T ltertg opertos tmes d c e coected to the rst etreme Stte 7

15 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 mtr Z L T L v ltertg T opertos or tmes. I the stte mtr A s the vlue o the ull pth P d Z equls to A. The stte mtr A s coected to the secod etreme stte mtr v Y tmes o " L T L T " ltertg opertos. It lso c rech the rst etreme stte mtr tmes o Test T L T T ltertg opertos. Gve the e ch stte mtr Z through A the 9-t r vlue o A equls to 7. From the Theorem It tes totl o 7 tmes o serl T d L ltertg opertos strtg wth T operto or the stte mtr A to rech the rst etreme stte mtr. Smlrl gve the other stte mtr o e ch rgs A the 9-t r vlue A equls to. B the Theorem rom the stte mtr A to cheve the rst etreme stte mtr t tes totl o tmes through seres o L d T ltertg opertos strtg wth L operto.. Iverse ucto Becuse s oe-to-oe d oto ucto ts verse ucto ets. Wht s ts verse ucto? The ollowgs re our dscusso: Oe-to-oe ucto stses the ollowg correspodece: M N M A 7 6 A M let A 7 6 the A 7 6 mog them XOR 7 Z Net let s corm whether elemets. A. holds. Tht mes we c derve elemets o Frst s determed the rst row the secod row ollows. So we c just d out the rst row s elemets. A rom o the e

16 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 9 Furthermore A so A s vector o rst colum s. Sce XOR dog XOR t operto o oth sdes o the equto t the sme tme we hve XOR XOR XOR. So XOR Gve A d XOR Z 7. Usg the grph elow we c llustrte ther correspodece: XOR XOR We wll dscuss the results ove geerlze them s Theorem 6. Theorem 6. I stte mtr A o the ull pth o the order rr grph s trsormed to -t r vlue through the stte mtr ucto let the A XOR Z. Test Gve two 9-t r umers d d pplg Theorem the correspodg

17 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 stte mtrces o the 9 order rr grph re Gve the e-rg stte mtr A the 9-t r vlue o A d t tes the L-operto ollowed seres o L d T opertos 7 tmes to rech the stte mtr A. From the theorem we c see tht the vlue o A s 7 7 epresso s d the Theorem A.. Its 9-t r.. To Eplore the Sequece the Perect Soluto Operto Order o the Bucles o -ch M Sword From the Theorem cocluso we ow tht the perect soluto o the -ch s rom the ull upper stte mtr Z the order rr grph to the rst etreme stte mtr whch t tes ] N X [ s tmes o T d L ltertg opertos. Net we umer the ucles rom outsde to sde o -rg m sword s.... We wll stud the order chge o ucle umers d ts relted propertes whle the [ ucles go through ] N tmes o serl T d L ltertg opertos. the ucle umer sequece the perect soluto o oe-rg s. the ucle umer sequece the perect soluto o two-rg s. the ucle umer sequece the perect soluto o three-rg s. the ucle umer sequece the perect soluto o our- rg s. the ucle umer sequece the perect soluto o ve-rg s. 6 6 the ucle umer sequece the perect soluto o s-rg s 6. From the ove sequeces we oud severl propertes:. The sequece s ormed the opertos order o umered ucles o the -rg m sword the perect soluto. Ths sequece c e seprted to severl su-seres comtos whch re. t Prove B Theorem s the perect soluto o the -ch m sword reles o ulocg the rst ucles through the sequece. Thus the m sword hs ol the d umered ucles closel coected whch the L operto wll uloc the umered ucle. V the t-rrgemet o the sequece the rst ucles re operted reversel wth ltertg T d L opertos levg the m sword wth ucles. Fll

18 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 we c uloc the the sequece ucles d complete the perect soluto o the chs. The recursve reltoshp o the -umered-ucles sequece s s ollowg: t N. Whe s odd: The rst rg s the m posto where the thrd rg s the m posto the th rg the m 6 posto the p rg the m p p posto d the rg the m posto m Z p p Z. The secod rg s the m posto where the ourth rg s the6m posto the sth rg the 6m posto the p rg the p p p m posto d the rg the m posto m Z p p N Proo B mthemtcl ducto. the umer ucle s the odd posto whch s the m the umer ucle s the m postos respectvel. the umer p p ucle s the ucle s the posto. It holds. posto d the secod d thrd ucles re the ourth d secod p p p m p p Z p m posto p p Z p. where the umer the recursve reltoshp o the -umered-ucles sequece the perect soluto o the chs o the -ch m sword s s ollow: t N We hve t t d. So the umered ucle s the susequece rom the m sequece. From Theorem the sequece hs terms the sequece hs terms d the sequece t hs terms.

19 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Thus the umered ucle s locted the umered term the sequece. Smlrl the umered ucle s locted the umered = = term o the sequece. So t lso holds. Thereore the mthemtcl ducto we c deduce tht or ll postve odd umer the sequece ormed the perect soluto o the ucles o the -ch m sword the umered ucle les the p p p p m posto p p N d the umered p ucle s the p m posto p p Z p c. Whe s eve: The rst rg s the posto the p. m posto where the thrd rg s the m 7 posto the th rg them 7 rg the p p m posto d the p rg the m posto. m Z p p Z The secod rg s the m posto where the ourth rg s the6m posto the sth rg the 6m posto the p rg the p p m posto d the rg the m posto. m Z p p Z Proo B mthemtcl ducto the umer d ucles re the eve d odd postos respectvel. So the umer d ucles re the m d m' postos respectvel m m' Z ucles re the seveth d thrd postos respectvel. I the umered p ucle s the ucle s the p p m posto p p Z p t holds.. The thrd d ourth p p m p p Z where the umered p. Whe the recursve reltoshp o the -umered-ucles sequece the perect soluto o the

20 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 chs o the -ch m sword s s ollow: We hve d t N t t. So the umered ucle s the sequece. From Theorem the sequece hs terms the sequece hs terms d the sequece t lso hs terms. Thus the umered ucle s locted the umered = term the susequece o the sequece. Smlrl the umered ucle s locted the umered = holds. term the sequece. So t lso Thereore the mthemtcl ducto we c deduce tht or ll postve eve umer the sequece ormed the perect soluto o the ucles o the -ch m sword the umered p ucle les the p p p m posto p p Z d the umered p p m posto p p N p Theorem 7.. p ucle s the The rgs o the -ch m sword re umered rom the outsde to sde s.... I the sequece s ormed the opertos order o umered ucles o the -rg m sword the perect soluto d the sequece the t s the reverse o the sequece The sequece stses the recursve reltoshp t N Whe s odd:

21 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 The rst rg s the m posto where the thrd rg s the m posto the th rg them 6 posto the p rg the m p p posto d the rg the m posto m Z. p p Z The secod rg s the m posto where the ourth rg s the6m posto the sth rg the 6m posto the p rg the p p p m posto d the rg the m posto m Z p p N Whe s eve:. The rst rg s the m posto where the thrd rg s the m 7 posto the th rg the m 7 posto the p rg the p p m posto d the p rg the m posto m Z p p Z. The secod rg s the m posto where the ourth rg s the6m posto the sth rg the 6m posto the p rg the p p m posto d the rg the m posto. m Z p p Z The ucle umerg sequece hs o-repettve propert. test the sequece the sequece ormed the opertos order o umered ucles o the -rg m sword the perect soluto there re 7umer ucles le the mold more th posto umer ucles the mold more th posto umer ucles the mold 6 more th posto umer ucles the mold 6 more th posto 6 umer ucles the mold more th 6 posto umer 6 ucles the mold o 6 more th

22 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 6 posto umer 7 ucles the mold more th posto umer ucles the module 6 more th posto d umer 9 ucles the mold more th 6 posto. Thereore ths sequece hs totl o tems. I the perect soluto o the e chs strtg rom the ll upper stte mtr Z ter seres o T L opertos ltertel whch ucle t s t the 6 th step o operto? Becuse 6 s dvded 6 more th the 6th step operted the umer ucle. I ddto strtg rom the stte mtr A t egs wth L operto d ter seres o T L ltertg opertos whch ucle t s t the 66th steps? B Theorem the 9-t r vlue o s eve umer. Ater the L opertos t moves towrd the rst etreme stte mtr. It wll rech the posto wth the vlue equlg to Tht s 66 7 stte mtr. Sce 7 s dvded more th the 66th step opertes the rst ucle. A d opertos egg rom the ll upper.. Grphcl Dscrmt Method or Stte-chgg Operto o Stte Mtrces Order Arr Grph Accordg to the Theorem results through the prt o vlue we c determe the seres o T d L opertos eeded or stte mtr A o the ull pth o the order rr grph to rech the rst or secod etreme stte mtr. We c eve determe the mmum umer o opertg steps requred to complete the ts. At the sme tme the result o Theorem 7 c e used to determe whch ucle s volved or gve stte mtr A movg towrd gve stte mtr A t the mthstep the perect soluto process o the chs. j Net we wt to eplore the grphcl dscrmt method o the stte chge operto the order rr grph v the mthemtcl model o the order rr grph d the results o the Theorem d Theorem 7. Through ths grphcl dgrm we c esl d qucl decdes wht s the rst step eeded or the curret e-rg stte to e solved the perect soluto wht s the e-rg stte ter movg m steps d lso whch ucle s volved t the mthstep? Costruct grphcl Dscrmt method. Frst we tr to d the gurtve terpretto ptter etwee the stte mtr o the 9 order rr grph d the 9-t r vlue v ther trsormto ucto. For emple s show elow the grph the 9 order rr o Fgure A s ts correspodg stte mtr the perect soluto o the e rgs. The ucto vlue s odd. From Theorem T operto s the et step we c A complete the perect soluto o the e rgs. I ddto the stte mtr A s oted seres o ltertg T

23 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 d L opertos tmes rom the ll upper stte mtr 9. Net step s the th step d s dvded more th. Thereore the et operto s T operto. B Theorem 7 t s o the rst ucle movg to other stte mtr A s show Fgure elow. Its ucto vlue s A j j eve. Smlrl L operto s et we c complete the e ch perect soluto. The stte mtr j A s oted sequetl d ltertg T d L opertos Z tmes rom the ll upper stte mtr Z 9. Net step s the th step d s dvded more th. From Theorem 7 L operto s the et operto. It s o the thrd ucle the movg to other stte mtr A s Fgure 6 elow. Fg. Fg. Fg. 6 O the other hd the perect soluto o the e rgs the purpose o ts m operto s to uloc the ermost rg o the m sword. So the rr dgrm o gure the purpose o seres o T L opertos s to uloc the ermost th lc crcle to F stte. To cheve ths gol we must rst move the 7th ucle to P stte provded tht the st to 6th ucles must e the F stte. Ever two ucles s med s secto so the rst step the perect soluto or gure s to move the rst ucle. Tht s the T operto. I ddto oservg the lc crcle stte the rst row o the grph gure we m to move the th lc crcle to F stte. To cheve ths the 7th ucle hs to e chged to P stte provded tht the rst to sth ucles re the F stte. From the sde to the outsde ever two ucles re treted s secto. I order to me the secod ucle to F stte the rst step the perect soluto or gure s to move the thrd ucle. Tht s the L operto. 6

24 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Smlrl oservg the lc crcle stte the rst row o the grph gure 6 we re to move the th lc crcle to F stte. To cheve ths the 7th ucle hs to e chged to P stte provded tht the rst to sth ucles re the F stte. From the sde to the outsde ever two ucles re treted s secto. So the rst step the perect soluto or gure 6 s to move the rst ucle. Tht s the T operto. Bsed o the ove dscusso we summrze the ollowg two prcples s the m prcples o the grphc dscrmt method: M prcple. The m operto the perect soluto s to uloc the ermost ucle o the m sword. So the trget ech stge o the operto s to move the ermost two ucles to djcet stte o the m sword. Complemetr prcple. To ollow the m prcple oe must chge the remg outsde ucle to F stte. Thus rom the sde out ever two ucles re treted s secto. I there s o ucle let we do the L operto. I there s ol oe ucle let secto we c the do the T operto. We summrze these vestgto results the ollowg grphc dscrmt method or the stte chge operto o the order rr grph. Grphc Dscrmt Method. The grphcl rule or the stte mtr chgg operto the order rr grph. I the umer o lc crcle the rst row s odd T operto rst c rech to the rst etreme stte mtr the shortest pth where L operto rst c rech to the secod etreme stte mtr the shortest pth. I the umer o lc crcle the rst row s eve L operto rst c rech to the rst etreme stte mtr shortest pth where T operto rst c rech to the secod etreme stte mtr shortest pth. test The ollowg gure s stte the perect soluto o e rgs. B grphc dscrmt method we c see tht strtg wth L operto movg the ourth ucle to P stte gves us the opportut to uloc the e rgs.. Ivestgto o the Geertg Fucto o Rgs We hve used the recursve reltoshp o the rgs to d the epresso o the geerl term. We wt to use the geerl term o the sequece to d the correspodet geertg ucto. Gve the sequece we c d the geertg ucto o the sequece. B Theorem the umer o tmes the rgs perect soluto opertos meets the ollowg recursve reltoshp: N The geerl ormul or s [ ] N. 7

25 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Wht s the geertg ucto or the sequece? Ivestgto: method. Frst let s the umer o tmes the perect soluto opertos or rgs. Becuse the recursve reltoshp or s N the sequece equls to d..... we hve Let the geerte ucto s multpled respectvel we hve 6 We dd them ll together method. Sce there s recursve reltoshp etwee d :

26 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 9 we multpl d respectvel Sutrcted we hve ` c method. Suppose tht s the geertg ucto or the umer o tmes the perect soluto opertos or rgs. The geerl ormul or s ] [ N. So we wt to merge some specl geertg uctos s ollows to the l orm : r r r r r. ] [ } { } { } {

27 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 6 We summrze the results o the ove dscussos Theorem. Theorem. Let the umer o tmes opertos or the perect soluto o the rgs e sequece. The geertg ucto o the sequece s. M Results.Thus 6 Mthemtcl model o order rr grph stses the ollowg mthemtcl ormul: Blc crcle the rst row o the rr grph represets the ucle o the e rg m sword whch s deed s the P stte. Blc crcle the secod row s the ucle uloced rom the m sword whch s deed s the F stte. We use -t r sstem to represet the ucle s stte o the e rg m sword. mes the ucle s o the m sword where mes the ucle s o the m sword. So the stte o e ucles c e cotuousl wrtte 9-t r umer 7 whch dctes the sttes o ech e ucles o the m sword the soluto process. Smlrl usg the mtr we c dcte the stte o the lc crcle o the 9 order rr grph. The vector s used to represet the lc crcle o the rst row d dctes tht the lc crcle s o the secod row. Thus the sttes o the e lc crcles o the 9 order rr grph c e represeted 9 mtr. So the sttes o the lc crcle o the rr grph the operto process c e descred s 9 stte mtr. It c e deduced tht there s oe-to-oe d surjecto correspodece etwee the smolc represettos o these two sttes. Theorem For ll postve tegers d ll stte mtrces o the order rr grph the two etreme stte mtrces re the ed pots the c e reched ether w through seres o T d L opertos ltertel. The ollowgs re the two results: There must e sgle ull pth the stte chge operto process o the order rr grph. The umer o opertos requred to complete ths ull pth s strtg d edg wth T operto. Theorem For ll postve tegers d ll stte mtrces o the order rr grph there s uque ull pth P there wll e the ollowg two results:

28 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 B Gve two stte mtrces A d there must e shortest pth through T d L opertos or the stte mtr A B. chgg to the stte mtr There s smmetr etwee the stte mtrces o the rst hl d those o the secod hl o the whole pth P whch the trsormtos etwee these stte mtrces re completed oe L operto. Theorem For ll postve tegers rgs perect soluto d the umer o steps to complete the soluto ll meet the ollowg three results: The umer o steps to complete the perect soluto meets the ollowg recursve reltoshp N The geerl ormul or s [ ] N Whe s odd the perect soluto process s T recursve steps recursvep steps Z Z X Z X Z X recursvep - steps recursvep- steps Z X Z X X Whe s eve the perect soluto process s LT recursvep steps recursvep steps Z Z X Z X Z X recursvep - seps recursvep- steps Z X Z X X Theorem I there s trsormto ucto etwee ll stte mtrces A d -t r vlues 6 o the ull pth o the order rr grph the A trsorms A to. A The r umer s d XOR Z. The stte mtr trsormto ucto s oe-to-oe d relected ucto d s strctl cresg ucto. Theorem Through the mtr trsormto ucto o the ull pth o the order rr grph rtrr stte

29 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 mtr A s trsormed to -t r vlue A.Let d e rst d secod etreme stte mtrces o the ull pth o the order rr grph. The: I A s the odd-umered stte mtr o the ull pth coected to the secod etreme stte mtr Y opertos d c e coected to the rst etreme stte mtr opertos. I A s the eve-umered stte mtr o the ull pth coected to the secod etreme stte mtr Z Y P the vlue A s eve. The stte mtr A v serl order o T Y d c e coected to the rst etreme stte mtr Theorem 6 Z s T L T ltertg L T L v ltertg T P the vlues A v serl order o T Z s odd. The stte mtr A s L T L ltertg opertos T L T opertos.. v ltertg T I stte mtr A o the ull pth o the order rr grph s trsormed to -t r vlue through the stte mtr ucto the A Theorem 7 XOR Z. We umer the chs o the m sword rom outsde to sde s.... I the umers o chs the perect soluto operto process orm sequece the The umer sequece stses recursve reltoshp t N where the sequece t s the reverse order sequece o. Whe s odd umer the umered p ucle rg s the the umered p ucle rg s the p m posto m Z p p Z. p p p p m posto m Z p p N.

30 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 Whe s eve umer the umered p ucle rg s the the umered p ucle rg s p p m posto m Z p p Z. p p p m posto where m Z p p Z. The ucle umers sequece s o-repettve. Grphc Judgmet: I the order rr grph the grphc determto rule or the stte mtr operto chge s ollows: I the umer o lc crcle the rst row o the order rr grph s odd T operto rst c me t to the rst etreme stte mtr shortest pth. L operto rst c me t to the secod etreme stte mtr shortest pth. I the umer o lc crcle the rst row o the order rr grph s eve L operto rst c me t to the rst etreme stte mtr shortest pth. T operto rst c me t to the secod etreme stte mtr shortest pth. Theorem. Let the umer o opertos the perect soluto or the rgs orm sequece ucto o the sequece s whch stses the ollowg mthemtcl ormul:. Dscusso 6 d the geertg We smpl the mges o the e rgs o the m sword to 9 rr grphs. Comed wth stte mtr represetto d r represetto o ech rr grphs we stud the correspodg stte mtr d ts r represetto to eplore the operto mode o ech stte mtr chge whether there s optml operto pth etwee two stte mtrces d whether such operto pth s helpul to uderstd the umer o steps d decde the pths the perect soluto o the e rgs. I dog so we tr to epd d geerlze the cses rgs. I ths stud the most dcult prolem s the use o e rgs equpmet to prctce hds-o epermets d reserch. The process ecouters dcultes epermetl operto d epresso. So we rst costruct set o mge-sed mthemtcl model d mthemtcl lguge o mtr smols. V the mge-sed mthemtcl models we c spec strctve descrpto d operle strctve prolems. I dog so we coverted the e rgs gme to rgorous mthemtcl prolem whch s oe o the most mportt cotrutos o our stud. There ws lso dcult eplorg d solvg the eteso o ths prolem: to decde the et operto mode whe the ucle s o deret posto o the m sword d whch ucle sttus should e chged. Ater oservg epermetg d summrzg we derved result Theorem 7. I the process o the perect soluto or the rgs through correspodg ucto vlues to the mtr epresso d ts r represetto o the curret ucles sttes we c decde whch ucles oe should operte the et step. Most mporttl the sequece ormed the ucle umers the process o perect soluto operto hs o-repettve propert. O the other hd Theorem the most trgug prt the oe-oto-oe mppg chrcterstc reverslt d cremet the trsormto ucto etwee the stte mtr d the r represetto o the rr grph were oud. There s somorphc reltoshp etwee the stte mtr d the r represetto o the rr grph whch there s lso reversle ucto etwee the stte mtr d the r represetto o the ull pth o the rr grph. Ths reversle ucto hs strctl cresg propert. Thereore the result o Theorem oe c determe the umer o perect soluto steps eeded or the chge etwee two stte mtrces the whole pth. Through Theorem our s

31 Jourl o Mthemtcs Reserch Vol. 9 No. 6; 7 coclusos rther th the recursve reltoshp o the sequece we clculte the umer o perect soluto steps to solve the e rgs to e steps d the deduce the umer o perect soluto steps or the rgs to e [. Future Outloo ] steps. Restructure d stud the e-rg to doule-deced -rgs s mportt uture drecto or eplorto. I ths cotet we ssume tht the rst thg to e eplored s the two m modes o opertos the perect soluto or the orgl e rg s stll vle or the rgs or there wll e thrd or ourth mode o opertos. We lso tr to th out the possle developmets o the prolems d lst them elow: I the operto o the doule-deced rgs c we d cert lw o opertos whch c meet our codtos? We c t help ut woder tht the stte mtrces o the rr grph whether we c d smlr three-t represetto or the mtrces to meet our the codtos? Is there perect soluto or the doule-deced rgs? Whether there re smlr lws o opertos to meet our proposed codtos? Is there somorphc reltoshp or stte mtr the rr grph so tht the correspodece etwee these stte mtrces d the -t represettos s oe-to-oe. Eplorg the ucle umers the perect soluto or the doule-deced rgs c we d sequece o o-repettve propert? These re worth o cotuous stud. Bre through o the operto modes the perect soluto or the doule-deced rgs wll d us towrds the ultmte stud the umer o tmes the perect soluto opertos or the doule-deced rgs. Reereces Grder M. 96. The Br Gr Code ch. Kotted Doughuts d Other Mthemtcl Etertmets. New Yor: W. H. Freem. Hsu C. F Iterestg Numers d Grphs Tpe Tw: The Commercl Press Ltd. Hsu C. N.. Mthemtcs Commo Hgh School volume Tchug Tw: LogTer Culture. Kuo C. G.. Ne Rg d the Gr s code Mthemtcs Trsmsso -. Press W. H. Fler B. P. Teuols S. A. & Vetterlg W. T. 99. Gr Codes Numercl Recpes FORTRAN: The Art o Scetc Computg d ed. pp. 6-. Cmrdge Egld: Cmrdge Uverst Press. Schwrtz J. T. & Mthemtcs. Itroducto to Mtrces d Vectors. Meol NY: Dover Pulcto Ic.. Rosee J. A. & Rosee C. P.. A mult-prdgm soluto or cet puzzle. Jourl o Computg Sceces Colleges 9-9. Wesste E. W. 7. Gr Code MthWorld--A Wolrm We Retreved rom Wu C. Y. Ne Rgs Retreved rom Mtrl-emples.com 9. Gr Code rom/to r d decml Retreved rom Youg Z. S.. Mthemtcs Commo Hgh School techer s mul volume. Tchug Tw: ChuHu Grphcs d Boos. Zhg W. & Rsmusse P. Ne Led Rgs Clsscl Chese Puzzles: Art d Igeut Retreved rom Coprghts Coprght or ths rtcle s reted the uthors wth rst pulcto rghts grted to the jourl. Ths s ope-ccess rtcle dstruted uder the terms d codtos o the Cretve Commos Attruto lcese