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[讨论] 这样的数组有多少?

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发表于 2024-3-23 12:11:37 | 显示全部楼层 |阅读模式

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a,b,c,d=正整数,  a<b<c<d<90, 满足  \(\tan(a^\circ)*\tan(b^\circ)*\tan(c^\circ)*\tan(d^\circ)\)=1, 这样的数组有多少?

补充内容 (2024-3-24 06:32):
不好意思,还得限制:a+d≠90。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-23 19:06:13 | 显示全部楼层
上面代码通过浮点计算找出了38组解,但是并没有证明它们的正切值乘积严格等于1。
如果我们记\(s_1(x)=1, c_1(x)=x\),于是\(\sin(x)=\sin(x) s_1(\cos(x)),\cos(x)=c_1(\cos(x))\)
假设已经存在整系数多项式\(s_n(x),c_n(x)\)使得\(\sin(nx)=\sin(x) s_n(\cos(x)), \cos(nx)=c_n(\cos(nx))\)
于是我们可以得到\(sin((n+1)x) = \sin(x) s_n(\cos(x))\cos(x)+c_n(x)\sin(x), \cos((n+1)x)=c_n(\cos(x))cos(x)-s_n(\cos(x))\sin^2(x)=c_n(\cos(x))cos(x)-s_n(\cos(x))(1-\cos^2(x))\)
于是我们可以得到\(s_{n+1}(x)=x*s_n(x)+c_n(x), c_{n+1}(x)=x*c_n(x)-s_n(x)(1-x^2)\). 所以我们知道\(s_n,c_n\)都是整系数多项式。
如果\(\tan(a\degree)\tan(b\degree)\tan(c\degree)\tan(d\degree)=1\),那么\(\sin(a\degree)\sin(b\degree)\sin(c\degree)\sin(d\degree)=\cos(a\degree)\cos(b\degree)\cos(c\degree)\cos(d\degree)\)
于是设\(u=\cos(1\degree),v=\sin(1\degree)\),那么必然有\(v^4 s_a(u)s_b(u)s_c(u)s_d(u)=c_a(u)c_b(u)c_c(u)c_d(u)\),故\((1-u^2)^2 s_a(u)s_b(u)s_c(u)s_d(u)-c_a(u)c_b(u)c_c(u)c_d(u)=0\), 我们可以得到一个以u为零点的整系数多项式,这个多项式必然是u的极小多项式的倍数。
另外\(c_{90}(u)=0\), 当然,对于1到180之间的任意一个奇数h都有\(c_{90}(\cos(h\degree))=0\),而如果h含因子3或5,显然\(\cos(h\degree)\)可以有次数更低的多项式作为极小多项式。
由此,我们可以知道u的极小多项式的次数很可能是1到180之间和90互素的整数数目。
计算\(c_{90}(x)\)并且因子分解发现它正好有一个48次不可约因式\(m(x)=281474976710656*x^48 - 3377699720527872*x^46 + 18999560927969280*x^44 - 66568831992070144*x^42 + 162828875980603392*x^40 - 295364007592722432*x^38 + 411985976135516160*x^36 - 452180272956309504*x^34 + 396366279591591936*x^32 - 280058255978266624*x^30 + 160303703377575936*x^28 - 74448984852135936*x^26 + 28011510450094080*x^24 - 8500299631165440*x^22 + 2064791072931840*x^20 - 397107008634880*x^18 + 59570604933120*x^16 - 6832518856704*x^14 + 583456329728*x^12 - 35782471680*x^10 + 1497954816*x^8 - 39625728*x^6 + 579456*x^4 - 3456*x^2 + 1\)
这个验证了这个多项式的48个根正好都是\(\cos(h\degree)\)其中h为和90互素的整数。
而如果一组a,b,c,d满足要求,那么必然\(m(x)|(1-x^2)^2 s_a(x)s_b(x)s_c(x)s_d(x)-c_a(x)c_b(x)c_c(x)c_d(x)\)。
于是,如果a,b,c,d满足要求,那么我们将它们都乘上一个和90互素的整数h以后,结果也会满足要求,只是乘上h以后这些角度可能会超过90度,而正切函数以180度为周期,大于180度的可以减去180的倍数,90度到180度之间的可以变换为用180度减这个角,是的正切值变号,由于最终乘积是正数1,所以变号的数目必然会正好是偶数个。
于是,这38组解中,如果任意两组解之间可以通过上面乘h的方法相互变换的,我们就认为它们等价,这样每个等价组中只需要验证一组即可。
我们可以分析出38组等价关系如下
  1. {0}:1 59 61 87 =>{0}
  2. {1}:2 58 62 84 =>{1}
  3. {2}:3 29 31 89 =>{0}
  4. {3}:3 39 75 81 =>{3}
  5. {4}:3 57 63 81 =>{4}
  6. {5}:3 63 69 75 =>{3}
  7. {6}:4 56 64 78 =>{6}
  8. {7}:5 55 65 75 =>{7}
  9. {8}:6 28 32 88 =>{6}
  10. {9}:6 42 66 78 =>{9}
  11. {10}:6 54 66 72 =>{10}
  12. {11}:7 53 67 69 =>{0}
  13. {12}:8 52 66 68 =>{6}
  14. {13}:9 15 51 87 =>{3}
  15. {14}:9 27 33 87 =>{4}
  16. {15}:9 33 69 75 =>{3}
  17. {16}:9 51 63 69 =>{4}
  18. {17}:10 50 60 70 =>{17}
  19. {18}:11 49 57 71 =>{0}
  20. {19}:12 24 48 84 =>{19}
  21. {20}:12 26 34 86 =>{1}
  22. {21}:12 48 54 72 =>{21}
  23. {22}:13 47 51 73 =>{0}
  24. {23}:14 46 48 74 =>{1}
  25. {24}:15 21 27 87 =>{3}
  26. {25}:15 21 57 81 =>{3}
  27. {26}:15 25 35 85 =>{7}
  28. {27}:15 51 57 63 =>{3}
  29. {28}:16 42 44 76 =>{6}
  30. {29}:17 39 43 77 =>{0}
  31. {30}:18 24 36 84 =>{21}
  32. {31}:18 36 42 78 =>{10}
  33. {32}:19 33 41 79 =>{0}
  34. {33}:20 30 40 80 =>{33}
  35. {34}:21 23 37 83 =>{0}
  36. {35}:21 27 39 81 =>{4}
  37. {36}:22 24 38 82 =>{1}
  38. {37}:27 33 39 75 =>{3}
复制代码

最后还余12组需要验证。
使用pari/gp计算上面过程验证全部通过,输出结果如下
  1. 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866234499642210130114844513307984847634033541120*x^46 + 44274207759490741465582314062122902094276984832*x^44 - 2059569263398812475563533327292398015403261952*x^42 + 86849306287901850305053803305243542508535808*x^40 - 3305155098778494497710796099851617315061760*x^38 + 112956929238765357970326253220615948337152*x^36 - 3447813247114566621671345474353925980160*x^34 + 93413659404757188742793010415140864000*x^32 - 2230988783959441127652434639604154368*x^30 + 46599448818030381503621870083112960*x^28 - 843610711669808165504470556344320*x^26 + 13099545239526595687381267906560*x^24 - 172362439384133479944405647360*x^22 + 1894363217785138986758963200*x^20 - 17093899257838802395955200*x^18 + 124010045901303852285952*x^16 - 704610371034844092416*x^14 + 3032415535152034816*x^12 - 9457971471265280*x^10 + 20125865365760*x^8 - 26846685616*x^6 + 20023176*x^4 - 7386*x^2 + 1
  2. Verify passed for [1, 59, 61, 87]
  3. Pol for [2, 58, 62, 84]is -803469022129495137770981046170581301261101496891396417650688*x^202 + 40575185617539504457434542831614355713685625593015519091359744*x^200 - 1009307742236295173378684252936407098377929936626261037397573632*x^198 + 16486205106226519904861372684270307905916890748762118100808105984*x^196 - 198896072967164643549179818179548979849792791798512674814673551360*x^194 + 1890118468029588473037662627481896676024350687949216134622083022848*x^192 - 14735209281373526463273206606083357760026570669318378845421137035264*x^190 + 96912337965956654816143012678471314498636291709747799329500555116544*x^188 - 548816068545614071636005759298379112840537608142379773393672924561408*x^186 + 2718013949144297669112688281153889647789709138770766839734640951951360*x^184 - 11916792408279530093015942682684084924528006005298080862961566423711744*x^182 + 46711444016366525502633410739288330031390001645231958565882903513792512*x^180 - 165026614715715948650750931361827850176687045286115537828152099913859072*x^178 + 529032206271199295705185067509742393942480875944684181401481487574892544*x^176 - 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  4. Verify passed for [2, 58, 62, 84]
  5. Pol for [3, 39, 75, 81]is -784637716923335095479473677900958302012794430558004314112*x^192 + 37662610412320084583014736539245998496614132666784207077376*x^190 - 889779170991061998273723150739686714482508884252776892203008*x^188 + 13792361788078384308338188310143045032780900500348599833460736*x^186 - 157777649311096144963403277843092479912895873077458550621470720*x^184 + 1420505082246852778932544163511393043750542737503792598001647616*x^182 - 10482589251652290377149612418098635673662719278416204248066818048*x^180 + 65202514611628725087212454075509274402589346554164112137589358592*x^178 - 348891036914737536106948977259340217277170508243120672046007189504*x^176 + 1631123851999683684288711065511183772334069771385567197786433126400*x^174 - 6744517883639351409953129188738688724560462672836310959858133237760*x^172 + 24908307382420969822895710158109280859715259685266396026035586990080*x^170 - 82825887858443803589327980653064773043937431800660134007870649794560*x^168 + 249652246428502294102082705758743583210217940115956244061540497489920*x^166 - 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  6. Verify passed for [3, 39, 75, 81]
  7. Pol for [3, 57, 63, 81]is -50216813883093446110686315385661331328818843555712276103168*x^198 + 2485732287213125582478972611590235900776532756007757667106816*x^196 - 60589724500819936072924957407512000081427985927689093135728640*x^194 + 969487378102435917283100963782933464550780619275442318456389632*x^192 - 11453943662942181043365295922631358251290665873089246360319819776*x^190 + 106557284180376611727266263363132661866281816451459802279555629056*x^188 - 812961780087960104575471517967511254464193476381840453329005576192*x^186 + 5230743240992278832917347594102853826872269544184579281797135728640*x^184 - 28968681975442765273354100031024883891191292778253808154058084581376*x^182 + 140253721580693059029222628339509365835335621208294671988680397160448*x^180 - 600927514538554569404461846422738266916823632357879565355755637833728*x^178 + 2301023611211922735009355242269663567997258794490441612311428916576256*x^176 - 7938119088858335600166682511414161950528177740849641762902330939801600*x^174 + 24839216300708276323931130660975998236465613966222028726287522448015360*x^172 - 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  8. Verify passed for [3, 57, 63, 81]
  9. Pol for [4, 56, 64, 78]is -3138550867693340381917894711603833208051177722232017256448*x^194 + 152219717083127008523017893512785910590482119528252836937728*x^192 - 3634245745359657328487052207617763615347760603737036481888256*x^190 + 56939687921407196625641380792126479680252217836037076817018880*x^188 - 658477522554273488345739599713157407775969398106486668611551232*x^186 + 5994235860077394374067962864055488466024102378001112769027375104*x^184 - 44733579317492469717113573926967288924637317214550857526305357824*x^182 + 281438802497486046107936469118058905988426757769620769410151088128*x^180 - 1523514485293850874596389958834653553586544041555305979569063854080*x^178 + 7207184290340550398653733039496185414308957443429650179024463331328*x^176 - 30160499475881651124800948045717732440314658866526253466569765027840*x^174 + 112753521513038929726621726039765919264574250024860069777779814563840*x^172 - 379613847841289034010920124345420697963559734630401141518321875681280*x^170 + 1158766027250343598293002037021242738258932339134235528667805725491200*x^168 - 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  10. Verify passed for [4, 56, 64, 78]
  11. Pol for [5, 55, 65, 75]is -196159429230833773869868419475239575503198607639501078528*x^190 + 9317572888464604258818749925073879836401933862876301230080*x^188 - 217798266267860124549888279498601941175895204044733541253120*x^186 + 3339767532209031589020345676268668803860318168974724222156800*x^184 - 37787853610397349349964072450039857514645374121544661965209600*x^182 + 336434452333434995212572020175462969256067003868087636339982336*x^180 - 2454691588356720005830858353725864327316684525504932890143621120*x^178 + 15093191487659410715009177067230109581662658598859956213283553280*x^176 - 79819762675121883588990840259390002595331367590124768435634176000*x^174 + 368743783923652384867372970148581623407714061878136246078799872000*x^172 - 1506318357328119992183218583056955931620511942772186565231897477120*x^170 + 5494734180553185293984747319861140669814284930909715594097564057600*x^168 - 18043101845692903395135307912690093941047469337734431262640596582400*x^166 + 53694106861530706648495880457255722842817273485500442447167094784000*x^164 - 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  12. Verify passed for [5, 55, 65, 75]
  13. 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126458681782552421004214272*x^20 + 1498232776802829763870720*x^18 - 14265185428011660410880*x^16 + 106363736743130431488*x^14 - 600972627636845568*x^12 + 2465473957917696*x^10 - 6936264876672*x^8 + 12369418496*x^6 - 12502632*x^4 + 6120*x^2 - 1
  14. Verify passed for [6, 42, 66, 78]
  15. Pol for [6, 54, 66, 72]is -12259964326927110866866776217202473468949912977468817408*x^186 + 570088341202110655309305094099915016306170953452300009472*x^184 - 13040770804998281240200354027535555998003660560221362716672*x^182 + 195623438915415929241657587602493754866227953631517363666944*x^180 - 2164487363274279272035098719969899993746519459480593907056640*x^178 + 18838215156718078106281165820820903094745260245965633048543232*x^176 - 134309496950675186498486089648445327619943059161051272660910080*x^174 + 806660909259621800474347220789605197928077806637263649169735680*x^172 - 4165293304624999367196541512967673469357553449581319952489512960*x^170 + 18780426340532993066471537612611622532115899341881877312464486400*x^168 - 74844267245737734868313275462987454795602794309067890607741992960*x^166 + 266231750631267085460142937004055374915787082613684353733253660672*x^164 - 852120109898830427265390444443535474407186337695044318007343972352*x^162 + 2470542097196675767445077097818618524176104275206997098962599542784*x^160 - 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1882653785328247428185325607554377087140648143925287255858180066902016*x^130 + 1733587962879129660150884605682347010287100229169068047481972310671360*x^128 - 1505406299389056183515639965276294497821960540885276184821952023625728*x^126 + 1233618585607577257563544716602582639994582649268631629081464327372800*x^124 - 954502367475992711665948864530366803226327849363086932160637881548800*x^122 + 697680924375153714082627111219908632859331140438596221655542503833600*x^120 - 481950638548625920912341096566384252962037958855609232064683966464000*x^118 + 314749787880904894234426735061281977123646928096525120997399566745600*x^116 - 194387137515336633754039474338773221056918982444798384912282880573440*x^114 + 113554104184317364230887742990752613534157007611731035481590244311040*x^112 - 62753583891333280232859015863310654847823609469640835397720924487680*x^110 + 32810721736311094976102372239712835846215117632173103803360975257600*x^108 - 16231194365788142703075300239816846362964978568381523439710592040960*x^106 + 7596854217460976799210600995339003365004214107068391752900710236160*x^104 - 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92890915123244407838246862210885025005344819674659107635200*x^74 + 17060952111239163985523240656772413139504317023296631603200*x^72 - 2936947305787129791688941725489931269600643461502656839680*x^70 + 473290253801943043302335466371670506129891547045848678400*x^68 - 71306560724922371206529518235551709661860371942822903808*x^66 + 10029676901964490900853270644106624956065575485355065344*x^64 - 1315025275387643968270448037558455780669805650975391744*x^62 + 160453787311784407278657952206045941935058399547883520*x^60 - 18186680554678432433858165847047324287358251082711040*x^58 + 1911162298593989117340620123860451897963565759332352*x^56 - 185807445696637830950097824317173866219568086646784*x^54 + 16674429624475651380223510693011560080615988527104*x^52 - 1377760074903708506869792754867186622042476642304*x^50 + 104530911170385425799255284971811000014602240000*x^48 - 7260521416268154072983320088366959184893181952*x^46 + 460173892580377234166791864735019835806711808*x^44 - 26518305456934884576061680471824235514298368*x^42 + 1383941144285524220456096523995708978626560*x^40 - 65122713111914452430850734609118705745920*x^38 + 2749625664730056021016546400505956401152*x^36 - 103604316315452990444452142919689699328*x^34 + 3462529575554547047370898817132003328*x^32 - 101935818482342596258509480016216064*x^30 + 2622860586616255266570579864453120*x^28 - 58457624148686592592112781361152*x^26 + 1116915236830646470479813017600*x^24 - 18073911193002256448107315200*x^22 + 244184987927943421124935680*x^20 - 2707470711756278953574400*x^18 + 24126137216265971515392*x^16 - 168326790255739846656*x^14 + 889335546753785856*x^12 - 3404992482725760*x^10 + 8897029191360*x^8 - 14590439368*x^6 + 13435116*x^4 - 6246*x^2 + 1
  16. Verify passed for [6, 54, 66, 72]
  17. Pol for [10, 50, 60, 70]is -187072209578355573530071658587684226515959365500928*x^170 + 7950568907080111875028045489976579626928273033789440*x^168 - 165968125935297335391210449603261099712127699580354560*x^166 + 2268396691301294419343939228858942874807972899953049600*x^164 - 22830866096335166573005340160337824145092895919256371200*x^162 + 180446863492307216532462206685433675561306942838195355648*x^160 - 1166302898181985667831767921259510342042593654929799249920*x^158 + 6339024910670976440892821002358328997525981201618886983680*x^156 - 29567442581023188723145912777203779930589935465884391833600*x^154 + 120198081796768180244093167159502322761311259393921332019200*x^152 - 431060370843659886400379120725765205002752504001450376953856*x^150 + 1377101098964522364357986470586171002260079826248201247129600*x^148 - 3950449829782720010413021315146721561863172033208843134566400*x^146 + 10243878355769253138221908530335141531680959326681921748992000*x^144 - 24146284695741810968665927250075690753247975555750244122624000*x^142 + 51984614212703498840282709182904890350702254470670041701416960*x^140 - 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629597058979414315403337742572213061244159702676700297072476160*x^110 + 437971033717984314323858176690695904172515365229382487926374400*x^108 - 286525886868491096938774065321426736221557266790689766218137600*x^106 + 176345560763454208633518875637440808125742777006214172639232000*x^104 - 102130928141118159757907341902479775986319021804204488392704000*x^102 + 55669461466126946128357747951288182656669765218355271927070720*x^100 - 28561757282061399599437277400381063863030196707178544365568000*x^98 + 13793337976187546463739624007723145350638338239099737997312000*x^96 - 6269699080085248392608920003510520613926517381408971816960000*x^94 + 2682017486695304435710862138906291407660286501677296517120000*x^92 - 1079512038394860035373622010909782446325770227597646210531328*x^90 + 408720525030385180721436580988250605717608135649613714554880*x^88 - 145515454782023295368457889883130310609781936123376115384320*x^86 + 48696116757763701205980002520197629241353639771312540876800*x^84 - 15309763980660557576122500793594822430569012292778301849600*x^82 + 4519442327090996596471362239121048623478245371855616606208*x^80 - 1251879185975416308279234295821977093627747701304233820160*x^78 + 325150769372442066576936605480993918991164322996069335040*x^76 - 79122241522494867635268979357177553955315723295260672000*x^74 + 18022362485432504425945631034811665417114005729705984000*x^72 - 3838763209397123442726656315279768053445994379892228096*x^70 + 763767766575552242065317616845226967196190318977351680*x^68 - 141775014036759127072164316385115376477637062237880320*x^66 + 24520835761074691789626090340447065297964407927603200*x^64 - 3945881616724663046995964458924683190536923145830400*x^62 + 589854714405244096914632634986915072593121821851648*x^60 - 81770272908621714802949557098715728604696380702720*x^58 + 10492645638716945863271038803576941300729910394880*x^56 - 1243740323554810486629890632730395321665427865600*x^54 + 135882760956400553493569946919149366488019763200*x^52 - 13650040986983896861893076233397305321199763456*x^50 + 1257429704395804063073524980487329806496563200*x^48 - 105912537825456655618724533742722108673228800*x^46 + 8130802303023582728179480368685560889344000*x^44 - 566902931800204729781909561095558791168000*x^42 + 35758492621338173335998170512665885016064*x^40 - 2031732535325326050391984353850310000640*x^38 + 103485579799751824060654270819556392960*x^36 - 4699820882726333438675586308793958400*x^34 + 189166273282189541497609864924364800*x^32 - 6701890842549589843228973802192896*x^30 + 207378186511460113980788016414720*x^28 - 5554773065756290559085568327680*x^26 + 127475039261756178427281408000*x^24 - 2476290096247622943768576000*x^22 + 40142045370993564578217984*x^20 - 533807693031450728529920*x^18 + 5702748728068200366080*x^16 - 47686541500274124800*x^14 + 301923176290585600*x^12 - 1385687970520832*x^10 + 4347146334080*x^8 - 8605439320*x^6 + 9696900*x^4 - 5550*x^2 + 1
  18. Verify passed for [10, 50, 60, 70]
  19. Pol for [12, 24, 48, 84]is -2787593149816327892691964784081045188247552*x^144 + 100353353393387804136910732226917626776911872*x^142 - 1768727853558460047913051655499423171943071744*x^140 + 20342461010784652796919613011831427261236510720*x^138 - 171694003513345824008286698036466501696847085568*x^136 + 1133921548383104075105087689190476608328817442816*x^134 - 6101675674904493003177195892739333971508509802496*x^132 + 27505259585622651984082396083745715076987787149312*x^130 - 105988787511693904819177799247338152996572883845120*x^128 + 354517228351871628958961791062701443438742551396352*x^126 - 1041725064653354133601426904708497711597051340390400*x^124 + 2715036002408229194410280784452154322843169351270400*x^122 - 6325656797277506379519118939667940885988961127759872*x^120 + 13260478429994542378448355594254159724952747675484160*x^118 - 25147550165453999171743716021248572047611677366026240*x^116 + 43342160104284333022826027776608233071023301119180800*x^114 - 68155970027273100392628423003790825780798714753843200*x^112 + 98114291266700304452692294266864473138962359618895872*x^110 - 129672657880121051208459919139398874820490255904276480*x^108 + 157736550996348855349354329119876380673671338003529728*x^106 - 176976594011849442698880665330655985387175788897370112*x^104 + 183485605408136773867346949911233367887674404442734592*x^102 - 176068247668218115355416486493663251226429515458674688*x^100 + 156582433706556528288360469036006272491378142766694400*x^98 - 129207692275082621849560052297149628795597447904624640*x^96 + 99023038140120833847404553746906057612464116782333952*x^94 - 70539390473457113509859527095638393197156780549865472*x^92 + 46735974807492049382968962745263374963955151566536704*x^90 - 28814282555308777708405143274419111137592970491985920*x^88 + 16536892658461299420181509099766399632306838425305088*x^86 - 8836599864764879027361239651621159387833995200823296*x^84 + 4396858050162381639554693301814807839659125402238976*x^82 - 2037104616699473570106272049842098043097449370746880*x^80 + 878685615994806941501303956079371280566263144775680*x^78 - 352766452914705503825911656729080005573546094886912*x^76 + 131767299462646635426043322877801249062377268183040*x^74 - 45769487765334247789746723692441557210570793418752*x^72 + 14774797115494842781966846599922351457742838824960*x^70 - 4429139927417619819162463456055679456923099332608*x^68 + 1231939940504486059851824697424090709612398903296*x^66 - 317609884118524961577728547401095342575027486720*x^64 + 75811354075310553171897767478782486790562381824*x^62 - 16731988832994704700467440825669197857298579456*x^60 + 3409596094170059296926384916486347965086040064*x^58 - 640466013791002381704169819049169013161590784*x^56 + 110698990768474615582729534484281691780153344*x^54 - 17570184956801598439293081494235016285126656*x^52 + 2555234643360647021335220723318780678111232*x^50 - 339656762140911665748744896600430021181440*x^48 + 41155082271372651746556412687457735147520*x^46 - 4531808068523505575435086182066281250816*x^44 + 451993711670535008278390447732472938496*x^42 - 40681103655045429082033781986528591872*x^40 + 3290453571249686908102819513054003200*x^38 - 238075716924389221982230495461638144*x^36 + 15329315788737163392103675758379008*x^34 - 873273653677399215869131298439168*x^32 + 43726120468998962970761824829440*x^30 - 1910069201420768104924638609408*x^28 + 72171328157214378192569106432*x^26 - 2335686050559092884808663040*x^24 + 64007644463345634197372928*x^22 - 1465480888145412243849216*x^20 + 27587006036936970207232*x^18 - 418765509537482317824*x^16 + 5004007208447016960*x^14 - 45644744670580736*x^12 + 305148916101120*x^10 - 1413261625536*x^8 + 4176429440*x^6 - 6917232*x^4 + 5040*x^2 - 1
  20. Verify passed for [12, 24, 48, 84]
  21. Pol for [12, 48, 54, 72]is -730750818665451459101842416358141509827966271488*x^162 + 29595408155950784093624617862504731148032633995264*x^160 - 588208737099521833860789280017281531567148600655872*x^158 + 7647330153297032814858877819763462092271849155067904*x^156 - 73145744899811017983895437374161595486998936934154240*x^154 + 548872624627180638769600094079616405370913678994440192*x^152 - 3364776837725173616730732200671836467968272104775745536*x^150 + 17327825420082403302749277577192521903016332152474173440*x^148 - 76498411087522882762705617485673804083202784559502458880*x^146 + 294021670792095030516789746218539294197785575367586611200*x^144 - 995804974498490274408180008903315977980394830258115706880*x^142 + 3000904154035034959994066848022605736171502903586392637440*x^140 - 8110777060766913933539519564238876059152423125526555656192*x^138 + 19791217231654526952439101301174521266692786733506073985024*x^136 - 43842513075914227177459206067756299524189136634552548065280*x^134 + 88589712929585414598262808451196459197290112596484751884288*x^132 - 163944062069605200046357072317860570680601522583306328080384*x^130 + 278837923012908641458154979997010321532828350276069687820288*x^128 - 437190323736288857594884968637287788082335808457541362384896*x^126 + 633580014615987256819970395292352103634227915274777001984000*x^124 - 850648097088295566991037717341457243365250366529128418508800*x^122 + 1060221817731967169868431170499374520985969766185663096422400*x^120 - 1228893470552961946892954311260638649324646774442473134489600*x^118 + 1326720617332983471911489460530526837079768014384797004595200*x^116 - 1335933954953351412688652581784211051226155292262469206016000*x^114 + 1256167971365625758996731573616365457072652005468987468021760*x^112 - 1104120490668655220294638498529316696997116072680320432865280*x^110 + 907983860296102612574260329037483346844679129728192180387840*x^108 - 699137892429488392251342774677315935243677506872267379834880*x^106 + 504368987732707143230324415488626287222072248605673835724800*x^104 - 341085896249542861002729491080439696257754919637877407088640*x^102 + 216317834471976763977475942939589371512397881193635376005120*x^100 - 128698711614938098760637730474875888099323258642847970099200*x^98 + 71846955168639979258786837320540885146356994166015551078400*x^96 - 37640408498276459721974721758003956372631881134769176576000*x^94 + 18506887013044590038779697278316680911673110510694721126400*x^92 - 8539443235957303119745570194778993619959015676697513033728*x^90 + 3697348125406378269684433365418535728436564785177036849152*x^88 - 1501851509343762310733447678604165211647751001994694754304*x^86 + 572156271279075593676334299322844402112498505651983482880*x^84 - 204358274762178024544641534797742976300895103674417676288*x^82 + 68400910147671156975685807768274228565165328157040967680*x^80 - 21443142466928656353488819624239056658989683719805075456*x^78 + 6292139615808750735736497076571570353122799107610509312*x^76 - 1726945098714843951930841161033005374987813255957708800*x^74 + 442970437951452375514411615043072727610011614699323392*x^72 - 106093744321955797584081328437595033292085296822747136*x^70 + 23701793944266720737187995540144176421776300890390528*x^68 - 4933542270197330943302943834891494101273638973997056*x^66 + 955612079607483734682816217261380546415068425748480*x^64 - 172010174329347072622930466447515256559068912812032*x^62 + 28729132631760759398265007170031070288994514239488*x^60 - 4444979436907040804966167556944748602486482272256*x^58 + 635931366557671822006057162965534055944696102912*x^56 - 83962131730008201646671750637419512475020165120*x^54 + 10208140229876151046400163580171483179273682944*x^52 - 1140161215028305434121576559107689260859260928*x^50 + 116683165280683921609789855267171435932549120*x^48 - 10910030705956054670508807063696977417994240*x^46 + 929067120345462698669534888986238622105600*x^44 - 71805350968223952310854009835239734509568*x^42 + 5017400355848762966114180350422357639168*x^40 - 315610667566660081706539171554879602688*x^38 + 17786940785161481819731785098624761856*x^36 - 893317339333830094193397278567301120*x^34 + 39742349639045484349168296602370048*x^32 - 1555571783436726970410350214643712*x^30 + 53155832830432036472980531838976*x^28 - 1571723051353688448509707026432*x^26 + 39801326561000669963721113600*x^24 - 852886897567872424306802688*x^22 + 15246704496160437202059264*x^20 - 223528225321647891382272*x^18 + 2632130050303785222144*x^16 - 24256824985682565120*x^14 + 169266650837480448*x^12 - 856698182226432*x^10 + 2970010872576*x^8 - 6535323952*x^6 + 8274120*x^4 - 5274*x^2 + 1
  22. Verify passed for [12, 48, 54, 72]
  23. Pol for [20, 30, 40, 80]is -170141183460469231731687303715884105728*x^130 + 5529588462465250031279837370766233436160*x^128 - 87782216841635844246567418260913955799040*x^126 + 907198107123205083256848318641335173120000*x^124 - 6863385679485676552496751267816291696640000*x^122 + 40526919760227022907182816886201639210319872*x^120 - 194463848849476440562691742316854639759196160*x^118 + 779549331363493536123264284095962588999843840*x^116 - 2663726455017675146486768839610691838436966400*x^114 + 7877438979232711486924370108325282888600780800*x^112 - 20402566956212885010410947793925874259486310400*x^110 + 46720163828306705404768663142310718233116672000*x^108 - 95321012217549215467999408097741965197574144000*x^106 + 174379169293853705385701592067368825546342400000*x^104 - 287553827200037672671163052731458837765160960000*x^102 + 429330366361120071113740888816421713281053360128*x^100 - 582561763607491901232280720347080205861598003200*x^98 + 720696528612532686443472132286371341353667788800*x^96 - 815073455326990818366895263560361431273570304000*x^94 + 844638709486243040080387865087662320661823488000*x^92 - 803558555677505607135871397768118451052153405440*x^90 + 702982094471820168048639329599615900878005862400*x^88 - 566291136111557662916471793358040861427472793600*x^86 + 420518921812188303329423047590352291064971264000*x^84 - 288114976653860210000135363388770277453201408000*x^82 + 182253311203578287033286195909711810258008014848*x^80 - 106494177532700527478144440773415641489038376960*x^78 + 57497673746080613829832141261621047224162058240*x^76 - 28688447123684908502918643128551139948101632000*x^74 + 13227641987672668661121851213833847076552704000*x^72 - 5634979164869772241311461094039500602227032064*x^70 + 2217088455693549101452254097934867619769221120*x^68 - 805251258029759877121078990446862103614586880*x^66 + 269801381513421365883086559046821802200268800*x^64 - 83321351231743541720936886533627506694553600*x^62 + 23693481525072062920382416562715890634719232*x^60 - 6196492048537491186266719043361832414740480*x^58 + 1488367823662227817687803946076160833617920*x^56 - 327822570675186625307237226793182586470400*x^54 + 66093126739663162580411460630734490828800*x^52 - 12172723106768305860324652904396789645312*x^50 + 2043383985562338213862753120410822246400*x^48 - 311846591182153854365506482531309977600*x^46 + 43144700632732534020264920233279488000*x^44 - 5394234846293467808306244706369536000*x^42 + 607302005688321703458356901760204800*x^40 - 61322994504881012598141750345728000*x^38 + 5528994272077739970147513270272000*x^36 - 442885385699804057343969198080000*x^34 + 31339856182460490079126159360000*x^32 - 1946586255582214147116476923904*x^30 + 105352626070891779801250529280*x^28 - 4926911265917821326270136320*x^26 + 197181542837300315553792000*x^24 - 6677889274843392180224000*x^22 + 188867361951399328612352*x^20 - 4391279333204453621760*x^18 + 82354253920253706240*x^16 - 1216916135761254400*x^14 + 13753959338316800*x^12 - 114361530639616*x^10 + 662913895040*x^8 - 2472174160*x^6 + 5182200*x^4 - 4650*x^2 + 1
  24. Verify passed for [20, 30, 40, 80]
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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-4-2 12:12:52 | 显示全部楼层
说实在,看不大明白需求。
不过由于解太多了,我们不妨多加一些约束。
要求$/_PAB, /_PAD, /_PDA,/_PDC,/_PCD,/_PCB,/_PBC,/_PBA$全部两两不等,而且要求四边形为凸四边形,这样的约束下有1092组不同的解
  1. 3 6 87 75 18 15 126 30
  2. 3 6 87 81 12 9 132 30
  3. 3 6 93 12 30 48 18 150
  4. 3 6 93 18 48 30 12 150
  5. 3 6 93 30 48 18 12 150
  6. 3 6 93 48 30 12 18 150
  7. 3 30 87 75 18 15 126 6
  8. 3 30 87 81 12 9 132 6
  9. 3 30 93 48 18 12 150 6
  10. 6 3 30 126 15 18 75 87
  11. 6 3 30 132 9 12 81 87
  12. 6 3 150 12 18 48 30 93
  13. 6 3 150 12 30 48 18 93
  14. 6 3 150 18 12 30 48 93
  15. 6 3 150 18 48 30 12 93
  16. 6 12 54 72 30 18 144 24
  17. 6 12 54 72 36 30 126 24
  18. 6 12 54 105 18 15 126 24
  19. 6 12 126 15 18 105 54 24
  20. 6 12 126 15 54 105 18 24
  21. 6 12 126 18 15 54 105 24
  22. 6 12 126 18 30 108 36 24
  23. 6 12 126 18 36 108 30 24
  24. 6 12 126 18 105 54 15 24
  25. 6 12 126 30 18 36 108 24
  26. 6 12 126 30 36 72 54 24
  27. 6 12 126 30 54 72 36 24
  28. 6 12 126 30 108 36 18 24
  29. 6 12 126 36 18 30 108 24
  30. 6 12 126 36 30 54 72 24
  31. 6 12 126 36 72 54 30 24
  32. 6 12 126 36 108 30 18 24
  33. 6 24 54 72 30 18 144 12
  34. 6 24 54 72 36 30 126 12
  35. 6 24 54 105 18 15 126 12
  36. 6 24 126 15 18 105 54 12
  37. 6 24 126 15 54 105 18 12
  38. 6 24 126 18 15 54 105 12
  39. 6 24 126 18 30 108 36 12
  40. 6 24 126 18 36 108 30 12
  41. 6 24 126 18 105 54 15 12
  42. 6 24 126 30 18 36 108 12
  43. 6 24 126 30 36 72 54 12
  44. 6 24 126 30 54 72 36 12
  45. 6 24 126 30 108 36 18 12
  46. 6 24 126 36 18 30 108 12
  47. 6 24 126 36 30 54 72 12
  48. 6 24 126 36 72 54 30 12
  49. 6 24 126 36 108 30 18 12
  50. 6 87 30 75 18 15 126 3
  51. 6 87 30 81 12 9 132 3
  52. 6 87 30 126 15 18 75 3
  53. 6 87 30 132 9 12 81 3
  54. 6 93 30 48 18 12 150 3
  55. 9 12 81 3 30 87 6 132
  56. 9 12 81 6 87 30 3 132
  57. 9 12 81 24 27 96 63 48
  58. 9 12 81 24 63 96 27 48
  59. 9 12 81 27 24 63 96 48
  60. 9 12 81 27 96 63 24 48
  61. 9 12 81 30 87 6 3 132
  62. 9 12 81 63 24 27 96 48
  63. 9 12 81 63 96 27 24 48
  64. 9 12 81 87 30 3 6 132
  65. 9 12 81 96 27 24 63 48
  66. 9 12 81 96 63 24 27 48
  67. 9 12 99 15 18 105 54 48
  68. 9 12 99 15 54 105 18 48
  69. 9 12 99 18 15 54 105 48
  70. 9 12 99 18 30 108 36 48
  71. 9 12 99 18 36 108 30 48
  72. 9 12 99 18 105 54 15 48
  73. 9 12 99 24 30 84 54 48
  74. 9 12 99 24 54 84 30 48
  75. 9 12 99 30 18 36 108 48
  76. 9 12 99 30 24 54 84 48
  77. 9 12 99 30 36 72 54 48
  78. 9 12 99 30 54 72 36 48
  79. 9 12 99 30 84 54 24 48
  80. 9 12 99 30 108 36 18 48
  81. 9 12 99 36 18 30 108 48
  82. 9 12 99 36 30 54 72 48
  83. 9 12 99 36 72 54 30 48
  84. 9 12 99 36 108 30 18 48
  85. 9 12 99 54 15 18 105 48
  86. 9 12 99 54 24 30 84 48
  87. 9 12 99 54 30 36 72 48
  88. 9 12 99 54 72 36 30 48
  89. 9 12 99 54 84 30 24 48
  90. 9 12 99 54 105 18 15 48
  91. 9 12 99 72 36 30 54 48
  92. 9 12 99 72 54 30 36 48
  93. 9 48 81 24 27 96 63 12
  94. 9 48 81 24 63 96 27 12
  95. 9 48 81 27 24 63 96 12
  96. 9 48 81 27 96 63 24 12
  97. 9 48 81 63 24 27 96 12
  98. 9 48 81 63 96 27 24 12
  99. 9 48 81 96 27 24 63 12
  100. 9 48 81 96 63 24 27 12
  101. 9 48 99 15 18 105 54 12
  102. 9 48 99 15 54 105 18 12
  103. 9 48 99 18 15 54 105 12
  104. 9 48 99 18 30 108 36 12
  105. 9 48 99 18 36 108 30 12
  106. 9 48 99 18 105 54 15 12
  107. 9 48 99 24 30 84 54 12
  108. 9 48 99 24 54 84 30 12
  109. 9 48 99 30 18 36 108 12
  110. 9 48 99 30 24 54 84 12
  111. 9 48 99 30 36 72 54 12
  112. 9 48 99 30 54 72 36 12
  113. 9 48 99 30 84 54 24 12
  114. 9 48 99 30 108 36 18 12
  115. 9 48 99 36 18 30 108 12
  116. 9 48 99 36 30 54 72 12
  117. 9 48 99 36 72 54 30 12
  118. 9 48 99 36 108 30 18 12
  119. 9 48 99 54 15 18 105 12
  120. 9 48 99 54 24 30 84 12
  121. 9 48 99 54 30 36 72 12
  122. 9 48 99 54 72 36 30 12
  123. 9 48 99 54 84 30 24 12
  124. 9 48 99 54 105 18 15 12
  125. 9 48 99 72 36 30 54 12
  126. 9 48 99 72 54 30 36 12
  127. 12 6 24 15 54 105 18 126
  128. 12 6 24 18 30 108 36 126
  129. 12 6 24 18 36 108 30 126
  130. 12 6 24 18 105 54 15 126
  131. 12 6 24 30 54 72 36 126
  132. 12 6 24 30 108 36 18 126
  133. 12 6 24 36 72 54 30 126
  134. 12 6 24 36 108 30 18 126
  135. 12 6 24 54 72 36 30 126
  136. 12 6 24 54 105 18 15 126
  137. 12 6 24 72 54 30 36 126
  138. 12 6 24 105 54 15 18 126
  139. 12 6 24 108 30 18 36 126
  140. 12 6 24 108 36 18 30 126
  141. 12 6 24 126 15 18 105 54
  142. 12 6 24 126 30 36 72 54
  143. 12 6 24 144 18 30 72 54
  144. 12 9 48 15 18 105 54 99
  145. 12 9 48 15 54 105 18 99
  146. 12 9 48 18 30 108 36 99
  147. 12 9 48 18 36 108 30 99
  148. 12 9 48 18 105 54 15 99
  149. 12 9 48 24 27 96 63 81
  150. 12 9 48 24 30 84 54 99
  151. 12 9 48 24 54 84 30 99
  152. 12 9 48 24 63 96 27 81
  153. 12 9 48 27 24 63 96 81
  154. 12 9 48 27 96 63 24 81
  155. 12 9 48 30 36 72 54 99
  156. 12 9 48 30 54 72 36 99
  157. 12 9 48 30 84 54 24 99
  158. 12 9 48 30 108 36 18 99
  159. 12 9 48 36 30 54 72 99
  160. 12 9 48 36 72 54 30 99
  161. 12 9 48 36 108 30 18 99
  162. 12 9 48 54 30 36 72 99
  163. 12 9 48 54 72 36 30 99
  164. 12 9 48 54 84 30 24 99
  165. 12 9 48 54 105 18 15 99
  166. 12 9 48 63 24 27 96 81
  167. 12 9 48 63 96 27 24 81
  168. 12 9 48 72 36 30 54 99
  169. 12 9 48 72 54 30 36 99
  170. 12 9 48 84 30 24 54 99
  171. 12 9 48 84 54 24 30 99
  172. 12 9 48 96 27 24 63 81
  173. 12 9 48 96 63 24 27 81
  174. 12 9 48 105 18 15 54 99
  175. 12 9 48 105 54 15 18 99
  176. 12 9 48 108 30 18 36 99
  177. 12 9 48 108 36 18 30 99
  178. 12 9 132 3 30 87 6 81
  179. 12 9 132 6 87 30 3 81
  180. 12 9 132 30 87 6 3 81
  181. 12 18 48 3 30 93 6 150
  182. 12 18 48 6 93 30 3 150
  183. 12 18 48 30 93 6 3 150
  184. 12 18 48 84 27 24 117 30
  185. 12 18 48 93 30 3 6 150
  186. 12 18 48 117 24 27 84 30
  187. 12 18 84 24 30 96 54 42
  188. 12 18 84 24 54 96 30 42
  189. 12 18 84 30 24 54 96 42
  190. 12 18 84 30 96 54 24 42
  191. 12 18 84 54 24 30 96 42
  192. 12 18 84 54 96 30 24 42
  193. 12 18 96 24 30 84 54 42
  194. 12 18 96 24 54 84 30 42
  195. 12 18 96 30 24 54 84 42
  196. 12 18 96 30 36 72 54 42
  197. 12 18 96 30 54 72 36 42
  198. 12 18 96 30 84 54 24 42
  199. 12 18 96 36 30 54 72 42
  200. 12 18 96 36 72 54 30 42
  201. 12 18 96 54 24 30 84 42
  202. 12 18 96 54 30 36 72 42
  203. 12 18 96 54 72 36 30 42
  204. 12 18 96 54 84 30 24 42
  205. 12 18 96 72 36 30 54 42
  206. 12 18 96 72 54 30 36 42
  207. 12 24 102 15 18 105 54 30
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  831. 36 30 54 48 99 12 9 72
  832. 36 30 54 78 48 18 24 72
  833. 36 30 54 96 42 12 18 72
  834. 36 30 54 99 48 9 12 72
  835. 36 30 126 12 54 24 6 72
  836. 36 30 126 24 54 12 6 72
  837. 36 72 54 9 48 99 12 30
  838. 36 72 54 12 42 96 18 30
  839. 36 72 54 12 99 48 9 30
  840. 36 72 54 12 126 24 6 30
  841. 36 72 54 18 48 78 24 30
  842. 36 72 54 18 96 42 12 30
  843. 36 72 54 24 78 48 18 30
  844. 36 72 54 24 126 12 6 30
  845. 36 72 54 42 96 18 12 30
  846. 36 72 54 48 78 24 18 30
  847. 36 72 54 48 99 12 9 30
  848. 36 72 54 78 48 18 24 30
  849. 36 72 54 96 42 12 18 30
  850. 36 72 54 99 48 9 12 30
  851. 36 108 30 9 48 99 12 18
  852. 36 108 30 12 99 48 9 18
  853. 36 108 30 12 126 24 6 18
  854. 36 108 30 24 126 12 6 18
  855. 36 108 30 48 99 12 9 18
  856. 36 108 30 99 48 9 12 18
  857. 39 30 51 15 54 75 18 78
  858. 39 30 51 18 75 54 15 78
  859. 39 30 51 54 75 18 15 78
  860. 39 30 51 75 54 15 18 78
  861. 39 78 51 15 54 75 18 30
  862. 39 78 51 18 75 54 15 30
  863. 39 78 51 54 75 18 15 30
  864. 39 78 51 75 54 15 18 30
  865. 42 12 18 24 54 84 30 96
  866. 42 12 18 24 54 96 30 84
  867. 42 12 18 30 54 72 36 96
  868. 42 12 18 30 84 54 24 96
  869. 42 12 18 30 96 54 24 84
  870. 42 12 18 36 72 54 30 96
  871. 42 12 18 54 72 36 30 96
  872. 42 12 18 54 84 30 24 96
  873. 42 12 18 54 96 30 24 84
  874. 42 12 18 72 54 30 36 96
  875. 42 12 18 84 54 24 30 96
  876. 42 12 18 96 54 24 30 84
  877. 42 21 30 24 63 84 27 69
  878. 42 21 30 27 84 63 24 69
  879. 42 21 30 63 84 27 24 69
  880. 42 21 30 84 63 24 27 69
  881. 42 69 30 24 63 84 27 21
  882. 42 69 30 27 84 63 24 21
  883. 42 69 30 63 84 27 24 21
  884. 42 69 30 84 63 24 27 21
  885. 42 84 18 24 54 96 30 12
  886. 42 84 18 30 96 54 24 12
  887. 42 84 18 54 96 30 24 12
  888. 42 84 18 96 54 24 30 12
  889. 42 96 18 24 54 84 30 12
  890. 42 96 18 30 54 72 36 12
  891. 42 96 18 30 84 54 24 12
  892. 42 96 18 36 72 54 30 12
  893. 42 96 18 54 72 36 30 12
  894. 42 96 18 54 84 30 24 12
  895. 42 96 18 72 54 30 36 12
  896. 42 96 18 84 54 24 30 12
  897. 48 9 12 15 54 105 18 99
  898. 48 9 12 18 105 54 15 99
  899. 48 9 12 24 54 84 30 99
  900. 48 9 12 24 63 96 27 81
  901. 48 9 12 27 96 63 24 81
  902. 48 9 12 30 54 72 36 99
  903. 48 9 12 30 84 54 24 99
  904. 48 9 12 30 108 36 18 99
  905. 48 9 12 36 72 54 30 99
  906. 48 9 12 36 108 30 18 99
  907. 48 9 12 54 72 36 30 99
  908. 48 9 12 54 84 30 24 99
  909. 48 9 12 54 105 18 15 99
  910. 48 9 12 63 96 27 24 81
  911. 48 9 12 72 54 30 36 99
  912. 48 9 12 84 54 24 30 99
  913. 48 9 12 96 63 24 27 81
  914. 48 9 12 105 54 15 18 99
  915. 48 18 24 30 54 72 36 78
  916. 48 18 24 36 72 54 30 78
  917. 48 18 24 54 72 36 30 78
  918. 48 18 24 72 54 30 36 78
  919. 48 24 30 15 54 105 18 66
  920. 48 24 30 18 105 54 15 66
  921. 48 24 30 54 105 18 15 66
  922. 48 24 30 105 54 15 18 66
  923. 48 66 30 15 54 105 18 24
  924. 48 66 30 18 105 54 15 24
  925. 48 66 30 54 105 18 15 24
  926. 48 66 30 105 54 15 18 24
  927. 48 78 24 30 54 72 36 18
  928. 48 78 24 36 72 54 30 18
  929. 48 78 24 54 72 36 30 18
  930. 48 78 24 72 54 30 36 18
  931. 48 81 12 24 63 96 27 9
  932. 48 81 12 27 96 63 24 9
  933. 48 81 12 63 96 27 24 9
  934. 48 81 12 96 63 24 27 9
  935. 48 99 12 15 54 105 18 9
  936. 48 99 12 18 105 54 15 9
  937. 48 99 12 24 54 84 30 9
  938. 48 99 12 30 54 72 36 9
  939. 48 99 12 30 84 54 24 9
  940. 48 99 12 30 108 36 18 9
  941. 48 99 12 36 72 54 30 9
  942. 48 99 12 36 108 30 18 9
  943. 48 99 12 54 72 36 30 9
  944. 48 99 12 54 84 30 24 9
  945. 48 99 12 54 105 18 15 9
  946. 48 99 12 72 54 30 36 9
  947. 48 99 12 84 54 24 30 9
  948. 48 99 12 105 54 15 18 9
  949. 51 30 39 15 54 75 18 78
  950. 51 30 39 18 75 54 15 78
  951. 51 30 39 54 75 18 15 78
  952. 51 30 39 75 54 15 18 78
  953. 51 78 39 15 54 75 18 30
  954. 51 78 39 18 75 54 15 30
  955. 51 78 39 54 75 18 15 30
  956. 51 78 39 75 54 15 18 30
  957. 54 15 18 12 99 48 9 105
  958. 54 15 18 12 126 24 6 105
  959. 54 15 18 24 63 84 27 75
  960. 54 15 18 24 102 30 12 105
  961. 54 15 18 24 126 12 6 105
  962. 54 15 18 27 84 63 24 75
  963. 54 15 18 30 66 48 24 105
  964. 54 15 18 30 102 24 12 105
  965. 54 15 18 39 78 51 30 75
  966. 54 15 18 48 66 30 24 105
  967. 54 15 18 48 99 12 9 105
  968. 54 15 18 51 78 39 30 75
  969. 54 15 18 63 84 27 24 75
  970. 54 15 18 84 63 24 27 75
  971. 54 24 30 12 99 48 9 84
  972. 54 24 30 18 84 42 12 96
  973. 54 24 30 18 96 42 12 84
  974. 54 24 30 42 84 18 12 96
  975. 54 24 30 42 96 18 12 84
  976. 54 24 30 48 99 12 9 84
  977. 54 30 36 12 99 48 9 72
  978. 54 30 36 12 126 24 6 72
  979. 54 30 36 18 96 42 12 72
  980. 54 30 36 24 78 48 18 72
  981. 54 30 36 24 126 12 6 72
  982. 54 30 36 42 96 18 12 72
  983. 54 30 36 48 78 24 18 72
  984. 54 30 36 48 99 12 9 72
  985. 54 72 36 12 99 48 9 30
  986. 54 72 36 12 126 24 6 30
  987. 54 72 36 18 96 42 12 30
  988. 54 72 36 24 78 48 18 30
  989. 54 72 36 24 126 12 6 30
  990. 54 72 36 42 96 18 12 30
  991. 54 72 36 48 78 24 18 30
  992. 54 72 36 48 99 12 9 30
  993. 54 75 18 24 63 84 27 15
  994. 54 75 18 27 84 63 24 15
  995. 54 75 18 39 78 51 30 15
  996. 54 75 18 51 78 39 30 15
  997. 54 75 18 63 84 27 24 15
  998. 54 75 18 84 63 24 27 15
  999. 54 84 30 12 99 48 9 24
  1000. 54 84 30 18 96 42 12 24
  1001. 54 84 30 42 96 18 12 24
  1002. 54 84 30 48 99 12 9 24
  1003. 54 96 30 18 84 42 12 24
  1004. 54 96 30 42 84 18 12 24
  1005. 54 105 18 12 99 48 9 15
  1006. 54 105 18 12 126 24 6 15
  1007. 54 105 18 24 102 30 12 15
  1008. 54 105 18 24 126 12 6 15
  1009. 54 105 18 30 66 48 24 15
  1010. 54 105 18 30 102 24 12 15
  1011. 54 105 18 48 66 30 24 15
  1012. 54 105 18 48 99 12 9 15
  1013. 63 24 27 12 81 48 9 96
  1014. 63 24 27 18 75 54 15 84
  1015. 63 24 27 30 69 42 21 84
  1016. 63 24 27 42 69 30 21 84
  1017. 63 24 27 48 81 12 9 96
  1018. 63 24 27 54 75 18 15 84
  1019. 63 84 27 18 75 54 15 24
  1020. 63 84 27 30 69 42 21 24
  1021. 63 84 27 42 69 30 21 24
  1022. 63 84 27 54 75 18 15 24
  1023. 63 96 27 12 81 48 9 24
  1024. 63 96 27 48 81 12 9 24
  1025. 66 30 24 18 105 54 15 48
  1026. 66 30 24 54 105 18 15 48
  1027. 66 48 24 18 105 54 15 30
  1028. 66 48 24 54 105 18 15 30
  1029. 69 30 21 27 84 63 24 42
  1030. 69 30 21 63 84 27 24 42
  1031. 69 42 21 27 84 63 24 30
  1032. 69 42 21 63 84 27 24 30
  1033. 72 36 30 12 99 48 9 54
  1034. 72 36 30 12 126 24 6 54
  1035. 72 36 30 18 96 42 12 54
  1036. 72 36 30 24 78 48 18 54
  1037. 72 36 30 24 126 12 6 54
  1038. 72 36 30 42 96 18 12 54
  1039. 72 36 30 48 78 24 18 54
  1040. 72 36 30 48 99 12 9 54
  1041. 72 54 30 12 99 48 9 36
  1042. 72 54 30 12 126 24 6 36
  1043. 72 54 30 18 96 42 12 36
  1044. 72 54 30 24 78 48 18 36
  1045. 72 54 30 24 126 12 6 36
  1046. 72 54 30 42 96 18 12 36
  1047. 72 54 30 48 78 24 18 36
  1048. 72 54 30 48 99 12 9 36
  1049. 75 18 15 27 84 63 24 54
  1050. 75 18 15 39 78 51 30 54
  1051. 75 18 15 51 78 39 30 54
  1052. 75 18 15 63 84 27 24 54
  1053. 75 54 15 27 84 63 24 18
  1054. 75 54 15 39 78 51 30 18
  1055. 75 54 15 51 78 39 30 18
  1056. 75 54 15 63 84 27 24 18
  1057. 81 12 9 27 96 63 24 48
  1058. 81 12 9 63 96 27 24 48
  1059. 81 48 9 27 96 63 24 12
  1060. 81 48 9 63 96 27 24 12
  1061. 84 18 12 30 96 54 24 42
  1062. 84 18 12 54 96 30 24 42
  1063. 84 30 24 12 99 48 9 54
  1064. 84 30 24 18 96 42 12 54
  1065. 84 30 24 42 96 18 12 54
  1066. 84 30 24 48 99 12 9 54
  1067. 84 42 12 30 96 54 24 18
  1068. 84 42 12 54 96 30 24 18
  1069. 84 54 24 12 99 48 9 30
  1070. 84 54 24 18 96 42 12 30
  1071. 84 54 24 42 96 18 12 30
  1072. 84 54 24 48 99 12 9 30
  1073. 99 12 9 18 105 54 15 48
  1074. 99 12 9 30 108 36 18 48
  1075. 99 12 9 36 108 30 18 48
  1076. 99 12 9 54 105 18 15 48
  1077. 99 48 9 18 105 54 15 12
  1078. 99 48 9 30 108 36 18 12
  1079. 99 48 9 36 108 30 18 12
  1080. 99 48 9 54 105 18 15 12
  1081. 102 24 12 18 105 54 15 30
  1082. 102 24 12 54 105 18 15 30
  1083. 102 30 12 18 105 54 15 24
  1084. 102 30 12 54 105 18 15 24
  1085. 105 18 15 12 126 24 6 54
  1086. 105 18 15 24 126 12 6 54
  1087. 105 54 15 12 126 24 6 18
  1088. 105 54 15 24 126 12 6 18
  1089. 108 30 18 12 126 24 6 36
  1090. 108 30 18 24 126 12 6 36
  1091. 108 36 18 12 126 24 6 30
  1092. 108 36 18 24 126 12 6 30
复制代码

比如随意选择了其中第790个作图,结果如下:
1.png

点评

{绿=棕}={21,27,63,69=24,30,42,84}={a1,a2,a3,a4=a5,a6,a7,a8}: a1<a2<a3<a4<90,a1<a5<a6<a7<a8<90,  发表于 2024-4-2 19:09
1个解可以有多个图:1092=577=310=42  发表于 2024-4-2 17:16
这是很难的题目, 这么多年没人敢动。谢谢mathe! 这是您的(我只是先睹为快)。  发表于 2024-4-2 15:40
要不再约束:8个角全部<90, 没有2个角的和=90。  发表于 2024-4-2 15:34
这些数据绝对可以媲美《三角形角格点问题》参考文献!  发表于 2024-4-2 15:22
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-28 12:44:12 | 显示全部楼层
[6]10,16,20,52,56,88
[6]10,26,40,46,62,82
[6]11,25,35,47,61,83
[6]11,27,39,49,71,75
[6]12,16,24,44,76,84
[6]12,16,44,54,72,76
[6]12,24,28,32,48,88
[6]13,21,27,47,73,81
[6]13,23,25,49,59,85
[6]13,23,49,55,59,75
[6]13,27,33,47,73,75
[6]1,35,37,71,73,85
[6]1,37,65,71,73,75
[6]1,39,59,61,75,81
[6]14,18,36,46,74,78
[6]14,20,22,50,58,86
[6]15,17,19,25,53,89
[6]15,17,43,57,63,77
[6]15,19,21,41,79,81
[6]15,19,41,51,63,79
[6]15,21,27,29,31,89
[6]15,31,35,41,67,77
[6]1,57,59,61,63,81
[6]1,59,61,63,69,75
[6]18,24,28,32,36,88
[6]18,26,34,36,42,86
[6]2,34,38,70,74,80
[6]2,42,58,62,66,78
[6]25,27,33,35,39,85
[6]2,54,58,62,66,72
[6]3,13,47,73,75,81
[6]3,19,41,63,79,81
[6]3,23,37,63,75,83
[6]3,25,35,39,81,85
[6]3,25,35,63,69,85
[6]4,24,48,56,64,84
[6]4,32,40,68,70,76
[6]4,48,54,56,64,72
[6]5,11,47,61,75,83
[6]5,17,19,53,55,89
[6]5,21,27,55,65,87
[6]5,21,55,57,65,81
[6]5,31,41,65,67,77
[6]5,51,55,57,63,65
[6]5,9,51,55,65,87
[6]6,14,46,66,74,78
[6]6,22,38,42,78,82
[6]6,22,38,54,72,82
[6]6,26,34,42,66,86
[6]7,15,27,53,67,87
[6]7,15,29,43,79,85
[6]7,15,53,57,67,81
[6]7,27,39,53,67,81
[6]7,29,43,55,65,79
[6]8,12,48,52,68,84
[6]8,18,36,52,68,84
[6]8,28,44,50,64,80
[6]9,11,27,49,71,87
[6]9,11,49,69,71,75
[6]9,15,17,43,77,87
[6]9,15,29,31,51,89
[6]9,17,43,63,69,77
[6]9,23,33,37,75,83
[6]9,23,37,51,63,83
[6]9,25,33,35,69,85
[6]9,27,29,31,33,89
[8]10,11,49,50,57,60,70,71
[8]10,12,16,20,26,52,86,88
[8]10,12,24,48,50,60,70,84
[8]10,12,26,34,50,60,70,86
[8]10,12,48,50,54,60,70,72
[8]10,13,47,50,51,60,70,73
[8]10,14,20,46,48,52,56,88
[8]10,14,46,48,50,60,70,74
[8]10,15,21,27,50,60,70,87
[8]10,15,21,50,57,60,70,81
[8]10,15,25,35,50,60,70,85
[8]10,15,50,51,57,60,63,70
[8]10,16,20,22,24,56,82,88
[8]10,16,20,52,56,58,62,84
[8]10,16,26,40,42,62,76,82
[8]10,16,42,44,50,60,70,76
[8]10,17,39,43,50,60,70,77
[8]10,18,24,36,50,60,70,84
[8]10,18,36,42,50,60,70,78
[8]10,19,33,41,50,60,70,79
[8]10,21,23,37,50,60,70,83
[8]10,21,27,39,50,60,70,81
[8]10,22,24,38,50,60,70,82
[8]10,26,40,46,52,62,66,68
[8]10,27,33,39,50,60,70,75
[8]1,10,50,59,60,61,70,87
[8]11,12,24,48,49,57,71,84
[8]11,12,26,34,49,57,71,86
[8]11,12,48,49,54,57,71,72
[8]11,13,25,27,47,49,53,89
[8]11,13,27,47,49,71,73,75
[8]11,13,47,49,51,57,71,73
[8]11,14,46,48,49,57,71,74
[8]1,11,49,57,59,61,71,87
[8]11,15,17,25,49,53,57,89
[8]11,15,21,27,49,57,71,87
[8]11,15,25,35,49,57,71,85
[8]11,15,31,35,57,67,71,77
[8]11,16,42,44,49,57,71,76
[8]11,17,25,27,39,49,53,89
[8]11,17,25,35,39,61,77,83
[8]11,17,39,43,49,57,71,77
[8]11,18,24,36,49,57,71,84
[8]11,18,36,42,49,57,71,78
[8]11,20,30,40,49,57,71,80
[8]11,21,23,37,49,57,71,83
[8]11,21,27,39,49,57,71,81
[8]11,22,24,38,49,57,71,82
[8]1,12,24,48,59,61,84,87
[8]1,12,26,34,59,61,86,87
[8]1,12,48,54,59,61,72,87
[8]11,25,27,35,39,49,71,85
[8]11,25,35,47,53,61,67,69
[8]11,27,31,35,39,67,71,77
[8]11,27,31,35,47,67,71,73
[8]1,13,47,51,59,61,73,87
[8]1,13,47,59,61,73,75,81
[8]1,14,46,48,59,61,74,87
[8]1,15,21,57,59,61,81,87
[8]1,15,25,35,59,61,85,87
[8]1,15,35,41,61,67,77,87
[8]1,15,51,57,59,61,63,87
[8]1,16,42,44,59,61,76,87
[8]1,17,39,43,59,61,77,87
[8]1,18,24,36,59,61,84,87
[8]1,18,36,42,59,61,78,87
[8]1,19,33,41,59,61,79,87
[8]1,19,41,59,61,63,79,81
[8]1,20,30,40,59,61,80,87
[8]1,21,23,37,59,61,83,87
[8]1,21,27,37,65,71,73,87
[8]1,21,27,39,59,61,81,87
[8]12,13,24,47,48,51,73,84
[8]12,13,26,34,47,51,73,86
[8]12,13,47,48,51,54,72,73
[8]1,21,37,41,65,73,79,81
[8]1,21,37,57,65,71,73,81
[8]12,14,26,34,46,48,74,86
[8]12,15,21,24,27,48,84,87
[8]12,15,21,24,48,57,81,84
[8]12,15,21,26,27,34,86,87
[8]12,15,21,26,34,57,81,86
[8]12,15,21,27,48,54,72,87
[8]12,15,21,48,54,57,72,81
[8]12,15,24,25,35,48,84,85
[8]12,15,24,48,51,57,63,84
[8]12,15,25,26,34,35,85,86
[8]12,15,25,35,48,54,72,85
[8]12,15,26,34,51,57,63,86
[8]12,15,48,51,54,57,63,72
[8]12,16,24,28,32,44,76,88
[8]12,16,26,34,42,44,76,86
[8]12,17,24,39,43,48,77,84
[8]12,17,26,34,39,43,77,86
[8]12,17,39,43,48,54,72,77
[8]12,18,24,26,34,36,84,86
[8]12,19,24,33,41,48,79,84
[8]12,19,26,33,34,41,79,86
[8]12,19,33,41,48,54,72,79
[8]12,20,24,30,40,48,80,84
[8]12,20,26,30,34,40,80,86
[8]12,20,30,40,48,54,72,80
[8]12,21,23,24,37,48,83,84
[8]12,21,23,26,34,37,83,86
[8]12,21,23,37,48,54,72,83
[8]12,21,24,27,39,48,81,84
[8]12,21,26,27,34,39,81,86
[8]12,21,27,39,48,54,72,81
[8]12,22,24,26,34,38,82,86
[8]12,22,24,38,48,54,72,82
[8]1,22,24,38,59,61,82,87
[8]12,24,27,33,39,48,75,84
[8]12,26,27,33,34,39,75,86
[8]12,26,32,34,40,68,70,76
[8]12,27,33,39,48,54,72,75
[8]1,23,37,43,59,63,79,85
[8]1,23,37,59,61,63,75,83
[8]1,25,35,39,59,61,81,85
[8]1,25,35,59,61,63,69,85
[8]1,2,58,59,61,62,84,87
[8]1,27,33,39,59,61,75,87
[8]13,14,46,47,48,51,73,74
[8]13,15,19,25,47,51,53,89
[8]13,15,21,27,47,51,73,87
[8]13,15,21,47,51,57,73,81
[8]13,15,25,35,47,51,73,85
[8]13,16,42,44,47,51,73,76
[8]13,18,24,36,47,51,73,84
[8]13,18,36,42,47,51,73,78
[8]13,19,21,23,55,59,79,81
[8]13,19,23,25,33,59,79,85
[8]13,19,23,33,55,59,75,79
[8]13,19,23,51,55,59,63,79
[8]13,19,25,27,33,47,53,89
[8]13,19,25,47,53,59,61,81
[8]13,19,33,41,47,51,73,79
[8]13,20,30,40,47,51,73,80
[8]13,21,23,27,29,49,55,89
[8]13,21,23,27,49,55,59,87
[8]13,21,23,37,47,51,73,83
[8]13,21,23,49,55,57,59,81
[8]13,22,24,38,47,51,73,82
[8]13,23,49,51,55,57,59,63
[8]13,25,27,33,35,47,73,85
[8]1,33,35,37,41,73,79,85
[8]1,33,37,41,65,73,75,79
[8]1,35,37,39,43,71,77,85
[8]1,35,37,41,61,63,77,83
[8]1,35,39,41,61,67,77,81
[8]1,35,41,47,61,67,73,81
[8]1,35,41,61,63,67,69,77
[8]1,37,39,43,65,71,75,77
[8]1,37,41,43,63,65,77,79
[8]1,37,41,51,63,65,73,79
[8]1,37,43,57,63,65,71,77
[8]1,37,51,57,63,65,71,73
[8]14,15,21,27,46,48,74,87
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[8]14,15,25,35,46,48,74,85
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[8]14,22,24,38,46,48,74,82
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[8]19,21,27,33,39,41,79,81
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[8]21,22,24,27,38,39,81,82
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[8]21,23,27,33,37,39,75,83
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[8]6,21,23,37,54,66,72,83
[8]6,21,27,28,32,39,81,88
[8]6,21,27,39,42,66,78,81
[8]6,21,27,39,54,66,72,81
[8]6,22,24,28,32,38,82,88
[8]6,22,26,34,38,42,82,86
[8]6,27,28,32,33,39,75,88
[8]6,27,33,39,42,66,75,78
[8]6,27,33,39,54,66,72,75
[8]6,28,32,34,38,70,74,80
[8]6,7,28,32,53,67,69,88
[8]6,7,42,53,66,67,69,78
[8]6,7,53,54,66,67,69,72
[8]6,8,28,32,52,66,68,88
[8]6,9,15,28,32,51,87,88
[8]6,9,15,42,51,66,78,87
[8]6,9,15,51,54,66,72,87
[8]6,9,27,28,32,33,87,88
[8]6,9,27,33,42,66,78,87
[8]6,9,27,33,54,66,72,87
[8]6,9,28,32,33,69,75,88
[8]6,9,28,32,51,63,69,88
[8]6,9,33,42,66,69,75,78
[8]6,9,33,54,66,69,72,75
[8]6,9,42,51,63,66,69,78
[8]6,9,51,54,63,66,69,72
[8]7,10,50,53,60,67,69,70
[8]7,11,49,53,57,67,69,71
[8]7,12,24,48,53,67,69,84
[8]7,12,26,34,53,67,69,86
[8]7,12,48,53,54,67,69,72
[8]7,13,15,29,51,73,79,85
[8]7,13,19,53,55,59,79,81
[8]7,13,25,49,53,59,69,85
[8]7,13,27,29,33,73,79,85
[8]7,13,27,29,49,53,55,89
[8]7,13,27,29,49,71,73,85
[8]7,13,27,47,53,67,73,81
[8]7,13,27,49,53,55,59,87
[8]7,13,29,51,55,65,73,79
[8]7,13,47,51,53,67,69,73
[8]7,13,49,53,55,57,59,81
[8]7,13,49,53,55,59,69,75
[8]7,14,46,48,53,67,69,74
[8]7,15,19,41,53,67,79,81
[8]7,15,25,35,53,67,69,85
[8]7,15,27,29,31,53,67,89
[8]7,15,29,43,49,57,71,85
[8]7,15,51,53,57,63,67,69
[8]7,16,42,44,53,67,69,76
[8]7,17,39,43,53,67,69,77
[8]7,18,24,36,53,67,69,84
[8]7,18,36,42,53,67,69,78
[8]7,19,33,41,53,67,69,79
[8]7,20,30,40,53,67,69,80
[8]7,22,24,38,53,67,69,82
[8]7,27,29,31,65,67,71,73
[8]7,27,29,33,39,43,79,85
[8]7,27,29,39,43,49,71,85
[8]7,27,33,39,53,67,69,75
[8]7,29,43,49,55,57,65,71
[8]7,8,52,53,66,67,68,69
[8]7,9,15,51,53,67,69,87
[8]7,9,27,33,53,67,69,87
[8]7,9,29,33,43,69,79,85
[8]7,9,29,43,49,69,71,85
[8]8,10,50,52,60,66,68,70
[8]8,11,49,52,57,66,68,71
[8]8,12,16,44,52,68,76,84
[8]8,12,26,34,52,66,68,86
[8]8,12,28,32,48,52,68,88
[8]8,12,28,34,44,50,80,86
[8]8,12,48,52,54,66,68,72
[8]8,13,47,51,52,66,68,73
[8]8,14,20,50,52,58,66,86
[8]8,14,28,48,50,64,74,80
[8]8,14,46,48,52,66,68,74
[8]8,15,21,27,52,66,68,87
[8]8,15,21,52,57,66,68,81
[8]8,15,25,35,52,66,68,85
[8]8,15,51,52,57,63,66,68
[8]8,16,42,44,52,66,68,76
[8]8,17,39,43,52,66,68,77
[8]8,18,28,32,36,52,68,88
[8]8,18,36,42,52,66,68,78
[8]8,19,33,41,52,66,68,79
[8]8,20,30,40,52,66,68,80
[8]8,21,23,37,52,66,68,83
[8]8,21,27,39,52,66,68,81
[8]8,27,33,39,52,66,68,75
[8]8,9,15,51,52,66,68,87
[8]8,9,27,33,52,66,68,87
[8]8,9,33,52,66,68,69,75
[8]8,9,51,52,63,66,68,69
[8]9,10,15,50,51,60,70,87
[8]9,10,27,33,50,60,70,87
[8]9,10,33,50,60,69,70,75
[8]9,10,50,51,60,63,69,70
[8]9,11,15,49,51,57,71,87
[8]9,11,17,23,25,49,83,89
[8]9,11,17,25,49,53,69,89
[8]9,11,23,37,49,71,75,83
[8]9,11,25,35,49,69,71,85
[8]9,11,27,29,31,49,71,89
[8]9,11,31,35,37,71,77,83
[8]9,11,31,35,67,69,71,77
[8]9,11,49,51,57,63,69,71
[8]9,12,15,24,48,51,84,87
[8]9,12,15,26,34,51,86,87
[8]9,12,15,48,51,54,72,87
[8]9,12,24,27,33,48,84,87
[8]9,12,24,33,48,69,75,84
[8]9,12,24,48,51,63,69,84
[8]9,12,26,27,33,34,86,87
[8]9,12,26,33,34,69,75,86
[8]9,12,26,34,51,63,69,86
[8]9,12,27,33,48,54,72,87
[8]9,12,33,48,54,69,72,75
[8]9,12,48,51,54,63,69,72
[8]9,13,23,29,49,51,55,89
[8]9,13,23,49,51,55,59,87
[8]9,13,27,33,47,51,73,87
[8]9,13,33,47,51,69,73,75
[8]9,14,15,46,48,51,74,87
[8]9,14,27,33,46,48,74,87
[8]9,14,33,46,48,69,74,75
[8]9,14,46,48,51,63,69,74
[8]9,15,16,42,44,51,76,87
[8]9,15,17,29,31,43,77,89
[8]9,15,18,24,36,51,84,87
[8]9,15,18,36,42,51,78,87
[8]9,15,19,33,41,51,79,87
[8]9,15,20,30,40,51,80,87
[8]9,15,21,23,37,51,83,87
[8]9,15,22,24,38,51,82,87
[8]9,15,25,27,33,35,85,87
[8]9,15,25,35,51,63,69,85
[8]9,16,27,33,42,44,76,87
[8]9,16,33,42,44,69,75,76
[8]9,16,42,44,51,63,69,76
[8]9,17,19,23,25,33,83,89
[8]9,17,19,25,33,53,69,89
[8]9,17,23,29,43,49,55,89
[8]9,17,23,37,43,63,77,83
[8]9,17,23,43,49,55,59,87
[8]9,17,27,33,39,43,77,87
[8]9,17,33,39,43,69,75,77
[8]9,18,24,27,33,36,84,87
[8]9,18,24,33,36,69,75,84
[8]9,18,24,36,51,63,69,84
[8]9,18,27,33,36,42,78,87
[8]9,18,33,36,42,69,75,78
[8]9,18,36,42,51,63,69,78
[8]9,19,33,41,51,63,69,79
[8]9,20,27,30,33,40,80,87
[8]9,20,30,33,40,69,75,80
[8]9,20,30,40,51,63,69,80
[8]9,21,23,27,33,37,83,87
[8]9,22,24,27,33,38,82,87
[8]9,22,24,33,38,69,75,82
[8]9,22,24,38,51,63,69,82
[8]9,23,25,33,35,37,83,85
[8]9,23,29,33,37,43,79,85
[8]9,23,29,37,43,49,71,85
[8]9,29,31,37,43,65,71,77
[8]9,29,31,37,51,65,71,73
[8]9,31,33,35,37,41,77,83
[8]9,31,33,35,41,67,69,77

评分

参与人数 1威望 +8 金币 +8 贡献 +8 经验 +8 鲜花 +8 收起 理由
northwolves + 8 + 8 + 8 + 8 + 8 很给力!

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-23 17:14:14 | 显示全部楼层
$C_44^2=946$:  {a,b,90-b,90-a},0,a<b<45

点评

不好意思,还得限制:a+d≠90。  发表于 2024-3-23 18:23
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-23 18:10:29 | 显示全部楼层
计算结果好像还有
  1. Special value 1 59 61 87 1.00000000000000
  2. Special value 2 58 62 84 1.00000000000000
  3. Special value 3 29 31 89 1.00000000000001
  4. Special value 3 39 75 81 1.00000000000000
  5. Special value 3 57 63 81 1.00000000000000
  6. Special value 3 63 69 75 1.00000000000000
  7. Special value 4 56 64 78 1.00000000000000
  8. Special value 5 55 65 75 1.00000000000000
  9. Special value 6 28 32 88 1.00000000000000
  10. Special value 6 42 66 78 1.00000000000000
  11. Special value 6 54 66 72 1.00000000000000
  12. Special value 7 53 67 69 1.00000000000000
  13. Special value 8 52 66 68 1.00000000000000
  14. Special value 9 15 51 87 1.00000000000000
  15. Special value 9 27 33 87 1.00000000000000
  16. Special value 9 33 69 75 1.00000000000000
  17. Special value 9 51 63 69 1.00000000000000
  18. Special value 10 50 60 70 1.00000000000000
  19. Special value 11 49 57 71 1.00000000000000
  20. Special value 12 24 48 84 1.00000000000000
  21. Special value 12 26 34 86 1.00000000000000
  22. Special value 12 48 54 72 1.00000000000000
  23. Special value 13 47 51 73 1.00000000000000
  24. Special value 14 46 48 74 1.00000000000000
  25. Special value 15 21 27 87 1.00000000000000
  26. Special value 15 21 57 81 1.00000000000000
  27. Special value 15 25 35 85 1.00000000000000
  28. Special value 15 51 57 63 1.00000000000000
  29. Special value 16 42 44 76 1.00000000000000
  30. Special value 17 39 43 77 1.00000000000000
  31. Special value 18 24 36 84 1.00000000000000
  32. Special value 18 36 42 78 1.00000000000000
  33. Special value 19 33 41 79 1.00000000000000
  34. Special value 20 30 40 80 1.00000000000000
  35. Special value 21 23 37 83 1.00000000000000
  36. Special value 21 27 39 81 1.00000000000000
  37. Special value 22 24 38 82 1.00000000000000
  38. Special value 27 33 39 75 1.00000000000000
复制代码

评分

参与人数 1威望 +3 金币 +3 贡献 +3 经验 +3 鲜花 +3 收起 理由
王守恩 + 3 + 3 + 3 + 3 + 3 对!就是这些数(我就是搞不出来)。.

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-3-23 20:41:52 | 显示全部楼层
mathe 发表于 2024-3-23 19:06
上面代码通过浮点计算找出了38组解,但是并没有证明它们的正切值乘积严格等于1。
如果我们记\(s_1(x)=1, c_ ...

38组解中的28组解(4,6,10,14,15,16,20,25,26,28不行)可以由恒等式推出来。\(\frac{\tan(a)*\tan(\pi/3+a)\ \ }{\tan(3a)*\tan(\pi/6+a)\ \ }=1\)  a好像可以是任意数。
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-23 23:50:51 | 显示全部楼层
  1. v=Select[Subsets[Range@89,{4}],N[Times@@Tan[# Degree]]==1&&#[[1]]+#[[4]]!=90&];{Length@v,v}
复制代码


{38,{{1,59,61,87},{2,58,62,84},{3,29,31,89},{3,39,75,81},{3,57,63,81},{3,63,69,75},{4,56,64,78},{5,55,65,75},{6,28,32,88},{6,42,66,78},{6,54,66,72},{7,53,67,69},{8,52,66,68},{9,15,51,87},{9,27,33,87},{9,33,69,75},{9,51,63,69},{10,50,60,70},{11,49,57,71},{12,24,48,84},{12,26,34,86},{12,48,54,72},{13,47,51,73},{14,46,48,74},{15,21,27,87},{15,21,57,81},{15,25,35,85},{15,51,57,63},{16,42,44,76},{17,39,43,77},{18,24,36,84},{18,36,42,78},{19,33,41,79},{20,30,40,80},{21,23,37,83},{21,27,39,81},{22,24,38,82},{27,33,39,75}}}

点评

用我们的方法(这里=4项), 怎么就推不出6项(4改6,或1=2项)来?  发表于 2024-3-27 13:26
没有45,应该有解吧?  发表于 2024-3-25 07:28
每个解加上45即可得到5个数的解  发表于 2024-3-24 19:46
4个数=38组解, 5个数的来几组试试?  发表于 2024-3-24 19:09
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2024-3-24 07:12:53 | 显示全部楼层
题目源自知乎《求证:tan15°=tan3°tan39°tan81°,这个三角函数问题怎么证明?》
也就是这里的第4道题:04--Special value 3 39 75 81 1.00000000000000
知乎作者作得辛苦, 知乎读者读得更辛苦。不知可有妙招,  把这38道题一并证明了。

又:连接网友 wayne帖子《[原创] 三角形的角格点问题》,里面有
四边形的4个角都是整数度数°,  四边形形内部存在一点P,
使得四边形的8个内角也都是整数度数°,每个四边形都可以有这样的P点。
我就特别好奇:这38道题都可以有这样的P点吗(四边形4个角都是90°;  P相关的4个角都是90°)?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-24 10:27:15 | 显示全部楼层
这个题目类似三角形角格点问题,我们可以将整数度数扩展到有理度数。
由于题目等价于\(\sin(a\degree)\sin(b\degree)\sin(c\degree)\sin(d\degree)-\cos(a\degree)\cos(b\degree)\cos(c\degree)\cos(d\degree)=0\)
我们分别记\(A=\exp(\frac{i a\pi}{180}),B=\exp(\frac{i b\pi}{180}),C=\exp(\frac{i c\pi}{180}),D=\exp(\frac{i d\pi}{180})\), 于是题目可以等价转化为
\((A-\frac1A)(B-\frac1B)(C-\frac1C)(D-\frac1D)=(A+\frac1A)(B+\frac1B)(C+\frac1C)(D+\frac1D)\)
展开后得到
\(A^2+B^2+C^2+D^2+B^2C^2D^2+A^2C^2D^2+A^2B^2D^2+A^2B^2C^2=0\)
于是我们这个方程要求8个单位根之和为0。
根据链接关联的论文 https://math.mit.edu/~poonen/papers/ngon.pdf 中TABLE 1结论,
权重之和不超过8的单位根之和为0的本源组合只有
权重 模式 模式变种数目
2 $R_2$ 1
3 $R_3$ 1
5$R_5$ 1
6 $(R_5:R_3)$1
7 $(R_5:2R_3)$2
7 $R_7$1
8 $(R_5:3R_3)$ 2
8 $(R_7:R_3)$ 1

由于不存在权重为1的方案,权重为7的不用考虑。所以我们只需要考虑权重分别为
8, 6+2, 5+3, 3+3+2, 2+2+2+2这5种组合





毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-24 11:02:42 | 显示全部楼层
对于权重为8的本源解,有两类,分别为
i) R7:R3, 也就是8个单位根角度为$x+\omega_7,x+2\omega_7,x+3\omega_7,x+4\omega_7,x+5\omega_7,x+6\omega_7,x-\omega_3,x-2\omega_3$
其中$\omega_7,\omega_3$分别为\(\frac{2\pi}7,\frac{2\pi}3\)
   而我们又要求这些角度分别为8个单位根$A^2,B^2,C^2,D^2,(BCD)^2,(ACD)^2,(ABD)^2,(ABC)^2$的幅角。
  注意到8个单位根两两乘积相等,也就是要求这8个角度两两相加为等角。显然这8个角是无法达成这个目标的,所以需要淘汰。
ii)R5:3R3, 也就是$x,x+\omega_5,x+2\omega_5,x+3\omega_5,x+4\omega_5$中淘汰三个角度,拼接上$\omega_3$
   由于表格已经给出本质上只有两种不等价组合,我们可以查看留下两个角度分别相邻和不相邻两种不同情况,对应
    $x-3\omega_5,x-4\omega_5, x+\omega_3,x+2\omega_3, x+\omega_5+\omega_3,x+\omega_5+2\omega_3, x+2\omega_5+\omega_3,x+2\omega_5+2\omega_3$ 以及
    $x-2\omega_5,x-4\omega_5, x+\omega_3,x+2\omega_3, x+\omega_5+\omega_3,x+\omega_5+2\omega_3, x+3\omega_5+\omega_3,x+3\omega_5+2\omega_3$。
其中第一种所有角度和为$8x-\omega_5+9\omega_3$,注意到$5\omega_5,3\omega_3$都是$2\pi$可以省略,所以8个角分成四组,每组内两个角的和都只能等于$2x-\frac{\omega_5}4 + h \frac{\pi}2$, 显然这个目标也无法达成,不符合要求。类似,第二种有不行。
所以权重为8的本源解都不符合本题要求。

现在我们再查看本源解为6+2类型的,其中权重6只能$R_5:R_3$,也就是6个角度可以为
$x+\omega_5,x+2\omega_5,x+3\omega_5,x+4\omega_5,x-\omega_3,x-2\omega_3$
而权重为2的为$R_2$,对应2两个角度$y,y+\omega_2$.
注意到前面6个角任意两个之间差值都不是$\pi$而后面两个差值为$\pi$,所以后面两个角如果不相互匹配,那么它们匹配对象差值应该为$\pi$,
得到矛盾,也就是后面两个角只能分到一组,也就是两个角度之和是$2y+\pi$。同时要求前面6个角分成3组,两两之和相等。
于是只能$x-\omega_3,x-2\omega_3$一组,$x+\omega_5,x+4\omega_5$一组,$x+2\omega_5,x+3\omega_5$,
  而且每组内两个角度之和为$2x=2y+\pi$, 得到$y=x+\frac{\pi}2$
得到这8个角度为$x+\omega_5, x+4\omega_5; x+2\omega_5,x+3\omega_5; x-\omega_3,x-2\omega_3; x+\frac{\pi}2,x+\frac{3\pi}2$.
但是另外一方面,比如$A^2$幅角为$x+\omega_5$,我们会要求$B^2C^2D^2$幅角为$x+4\omega_5$, 也就是后面三组各条选择一个,要求和为$x+4\omega_5$等等,这样下去好像有会无解,但是计算量有些大,可能还是需要借助计算机分析,枚举各种不同的排列情况。

通过程序解放长可以发现有解的有
5+3 R5 R3:

  1. Solution candiate with MOD 15:
  2. 2a = +1V1+0
  3. 2b = +1V1+3
  4. 2c = +1V1+6
  5. 2d = +1V2+0
  6. Constrain:
  7. +1V0-2V1+3 Mod 15=0
  8. +1V1-1V2+1 Mod 15=0
  9. +2V2+11 Mod 15=0
  10. Solution candiate with MOD 15:
  11. 2a = +1V1+0
  12. 2b = +1V1+3
  13. 2c = +1V1+6
  14. 2d = +1V2+0
  15. Constrain:
  16. +1V0-2V1+3 Mod 15=0
  17. +1V1-1V2+11 Mod 15=0
  18. +2V2+1 Mod 15=0
  19. Solution candiate with MOD 15:
  20. 2a = +1V1+0
  21. 2b = +1V1+3
  22. 2c = +1V1+6
  23. 2d = +1V2+0
  24. Constrain:
  25. +1V0-1V1-1V2+10 Mod 15=0
  26. +1V1-1V2+10 Mod 15=0
  27. +2V2+14 Mod 15=0
  28. ...
复制代码



3+3+2 R3 R3 R2

  1. Solution candiate with MOD 6:
  2. 2a = +1V1+0
  3. 2b = +1V1+2
  4. 2c = +1V1+4
  5. 2d = +1V2+0
  6. Constrain:
  7. +1V0-1V1-1V2+4 Mod 6=0
  8. +1V1-1V3+5 Mod 6=0
  9. +1V2-1V3=0
  10. +2V3=0
  11. Solution candiate with MOD 6:
  12. 2a = +1V1+0
  13. 2b = +1V1+2
  14. 2c = +1V1+4
  15. 2d = +1V2+0
  16. Constrain:
  17. +1V0-1V1-1V2+4 Mod 6=0
  18. +1V1-1V3+2 Mod 6=0
  19. +1V2-1V3+3 Mod 6=0
  20. +2V3=0
  21. Solution candiate with MOD 6:
  22. 2a = +1V1+0
  23. 2b = +1V1+2
  24. 2c = +1V1+4
  25. 2d = +1V2+0
  26. Constrain:
  27. +1V0-1V1-1V2+2 Mod 6=0
  28. +1V1-1V3+1 Mod 6=0
  29. +1V2-1V3=0
  30. +2V3=0
  31. ...
复制代码


2+2+2+2 R2 R2 R2 R2:

  1. Solution candiate with MOD 2:
  2. 2a = +1V1+0
  3. 2b = +1V1+1
  4. 2c = +1V2+0
  5. 2d = +1V2+1
  6. Constrain:
  7. +1V0-1V1-1V3=0
  8. +1V1-1V2+1V3-1V4=0
  9. +3V2-2V3+1V4=0
  10. Solution candiate with MOD 2:
  11. 2a = +1V1+0
  12. 2b = +1V1+1
  13. 2c = +1V2+0
  14. 2d = +1V2+1
  15. Constrain:
  16. +1V0-1V1-1V3=0
  17. +1V1-1V2+1V3-1V4+1 Mod 2=0
  18. +3V2-2V3+1V4+1 Mod 2=0
  19. Solution candiate with MOD 2:
  20. 2a = +1V1+0
  21. 2b = +1V1+1
  22. 2c = +1V2+0
  23. 2d = +1V2+1
  24. Constrain:
  25. +1V0-1V1-1V3=0
  26. +1V1-1V2+1 Mod 2=0
  27. +3V2-1V3+1 Mod 2=0
  28. +1V3-1V4+1 Mod 2=0
  29. ....
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现在代码给出解的方案还是太多了,主要原因在于有些等价的方案会被重复求解。
比如对于最后一种2+2+2+2, 我们会引入约束变量V1,V2,V3,V4等,这四个变量之间本质上是等价的,可以任意置换,得出看似不同的解,代码没有考虑。
另外V1和V1+1Mod2也可以相互置换等。类似3+3+2模式V1,V2可以置换,V3,V3+3可以置换,V1,V1+2,V1+4可以轮换,V2,V2+2,V2+4可以轮换等。
另外需要注意上面方程中常数项使用1Mod6之类的表示,代表将一个周角(360度)均分为6分,所以1Mod6实际上代表60度角。

比如我们解读第一种中第一个解:
Solution candiate with MOD 15:
2a = +1V1+0
2b = +1V1+3
2c = +1V1+6
2d = +1V2+0
Constrain:
+1V0-2V1+3 Mod 15=0
+1V1-1V2+1 Mod 15=0
+2V2+11 Mod 15=0
于是可以V2=2 Mod 15, V1=1 Mod 15, 代入得到2a=1 Mod 15, 2b=4 Mod 15, 2c=7 Mod 15, 2d = 2Mod15
对应$a=\frac{\pi}{15},b=\frac{4\pi}{15},c=\frac{7\pi}{15},d=\frac{2\pi}{15}$.
由此我们得到$\tan(\frac{\pi}{15})\tan(\frac{4\pi}{15})\tan(\frac{7\pi}{15})\tan(\frac{2\pi}{15})=1$
另外需要注意表达式中同余仅适用于常数项,非常数项是可以为分数的。准确的说,我们可以将上面表达式改为
2a = +1V1+0
2b = +1V1+3*2\pi/15
2c = +1V1+6*2\pi/15
2d = +1V2+0
Constrain:
+1V0-2V1+3*2\pi/15=整数*2\pi
+1V1-1V2+1*2\pi/15 =整数*2\pi
+2V2+11*2\pi/15=整数*2\pi
比如可以选择V2=$\frac{19\pi}{15}$,可以得到$\tan(\frac{17\pi}{30})\tan(\frac{23\pi}{30})\tan(\frac{29\pi}{30})\tan(\frac{19\pi}{30})=1$

进一步更新程序,将不含可变参数的方程全部解出,存放在附件中sv.e?u文件中,然后余下至少含有一个可变参数的放入sv.o?文件。

比如sv.o5中第一个方程对应恒等式\(\tan(x-\frac{3\pi}6)\tan(x-\frac{\pi}6)\tan(x+\frac{\pi}6)\tan(3x)=1\)

sv.tgz (15.3 KB, 下载次数: 4) (附件中是程序找到的候选解)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2024-3-24 11:07:10 | 显示全部楼层
$\tan (x) \tan \left(\frac{\pi }{3}-x\right) \tan \left(x+\frac{\pi }{3}\right)=\tan (3 x)$

  1. s = Table[Sort@{x, 60 - x, 60 + x, 90 - 3 x}, {x, 29}]
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{{1,59,61,87},{2,58,62,84},{3,57,63,81},{4,56,64,78},{5,55,65,75},{6,54,66,72},{7,53,67,69},{8,52,66,68},{9,51,63,69},{10,50,60,70},{11,49,57,71},{12,48,54,72},{13,47,51,73},{14,46,48,74},{15,45,45,75},{16,42,44,76},{17,39,43,77},{18,36,42,78},{19,33,41,79},{20,30,40,80},{21,27,39,81},{22,24,38,82},{21,23,37,83},{18,24,36,84},{15,25,35,85},{12,26,34,86},{9,27,33,87},{6,28,32,88},{3,29,31,89}}

点评

错啦!应是:s = Table[Sort@{ 90 - x, 30 + x, 30-x, 3 x}, {x, 29}]  发表于 2024-3-25 07:26
同理:s = Table[Sort@{3 x, 30 - x, 30 + x, 90 - x, 30-x, 3 x}, {x, 29}]  发表于 2024-3-25 07:22
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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