山水娱乐

  • REVERSAL-ADDITION PALINDROME TEST ON 18317699990

    Reverse and Add Process:

    1. Pick a number.
    2. Reverse its digits and add this value to the original number.
    3. If this is not a palindrome, go back to step 2 and repeat.
    Let's view this Reverse and Add sequence starting with 18317699990:
    18317699990
    + 09999671381
    step 1: 28317371371
    + 17317371382
    step 2: 45634742753
    + 35724743654
    step 3: 81359486407
    + 70468495318
    step 4: 151827981725
    + 527189728151
    step 5: 679017709876
    + 678907710976
    step 6: 1357925420852
    + 2580245297531
    step 7: 3938170718383
    + 3838170718393
    step 8: 7776341436776
    + 6776341436777
    step 9: 14552682873553
    + 35537828625541
    step 10: 50090511499094
    + 49099411509005
    step 11: 99189923008099
    + 99080032998199
    step 12: 198269956006298
    + 892600659962891
    step 13: 1090870615969189
    + 9819695160780901
    step 14: 10910565776750090
    + 09005767756501901
    step 15: 19916333533251991
    + 19915233533361991
    step 16: 39831567066613982
    + 28931666076513893
    step 17: 68763233143127875
    + 57872134133236786
    step 18: 126635367276364661
    + 166463672763536621
    step 19: 293099040039901282
    + 282109930040990392
    step 20: 575208970080891674
    + 476198080079802575
    step 21: 1051407050160694249
    + 9424960610507041501
    step 22: 10476367660667735750
    + 05753776606676367401
    step 23: 16230144267344103151
    + 15130144376244103261
    step 24: 31360288643588206412
    + 21460288534688206313
    step 25: 52820577178276412725
    + 52721467287177502825
    step 26: 105542044465453915550
    + 055519354564440245501
    step 27: 161061399029894161051
    + 150161498920993160161
    step 28: 311222897950887321212
    + 212123788059798222113
    step 29: 523346686010685543325
    + 523345586010686643325
    step 30: 1046692272021372186650
    + 0566812731202722966401
    step 31: 1613505003224095153051
    + 1503515904223005053161
    step 32: 3117020907447100206212
    + 2126020017447090207113
    step 33: 5243040924894190413325
    + 5233140914984290403425
    step 34: 10476181839878480816750
    + 05761808487893818167401
    step 35: 16237990327772298984151
    + 15148989227772309973261
    step 36: 31386979555544608957412
    + 21475980644555597968313
    step 37: 52862960200100206925725
    + 52752960200100206926825
    step 38: 105615920400200413852550
    + 055258314002004029516501
    step 39: 160874234402204443369051
    + 150963344402204432478061
    step 40: 311837578804408875847112
    + 211748578804408875738113
    step 41: 523586157608817751585225
    + 522585157718806751685325
    step 42: 1046171315327624503270550
    + 0550723054267235131716401
    step 43: 1596894369594859634986951
    + 1596894369584959634986951
    step 44: 3193788739179819269973902
    + 2093799629189719378873913
    step 45: 5287588368369538648847815
    + 5187488468359638638857825
    step 46: 10475076836729177287705640
    + 04650778277192763867057401
    step 47: 15125855113921941154763041
    + 14036745114912931155852151
    step 48: 29162600228834872310615192
    + 29151601327843882200626192
    step 49: 58314201556678754511241384
    + 48314211545787665510241385
    step 50: 106628413102466420021482769
    + 967284120024664201314826601
    step 51: 1073912533127130621336309370
    + 0739036331260317213352193701
    step 52: 1812948864387447834688503071
    + 1703058864387447834688492181
    step 53: 3516007728774895669376995252
    + 2525996739665984778277006153
    step 54: 6042004468440880447654001405
    + 5041004567440880448644002406
    step 55: 11083009035881760896298003811
    + 11830089269806718853090038011
    step 56: 22913098305688479749388041822
    + 22814088394797488650389031922
    step 57: 45727186700485968399777073744
    + 44737077799386958400768172754
    step 58: 90464264499872926800545246498
    + 89464254500862927899446246409
    step 59: 179928519000735854699991492907
    + 709294199996458537000915829971
    step 60: 889222718997194391700907322878
    + 878223709007193491799817222988
    step 61: 1767446428004387883500724545866
    + 6685454270053887834008246447671
    step 62: 8452900698058275717508970993537
    + 7353990798057175728508960092548
    step 63: 15806891496115451446017931086085
    + 58068013971064415451169419860851
    step 64: 73874905467179866897187350946936
    + 63964905378179866897176450947837
    step 65: 137839810845359733794363801894773
    + 377498108363497337953548018938731
    step 66: 515337919208857071747911820833504
    + 405338028119747170758802919733515
    step 67: 920675947328604242506714740567019
    + 910765047417605242406823749576029
    step 68: 1831440994746209484913538490143048
    + 8403410948353194849026474990441381
    step 69: 10234851943099404333940013480584429
    + 92448508431004933340499034915843201
    step 70: 102683360374104337674439048396427630
    + 036724693840934476733401473063386201
    step 71: 139408054215038814407840521459813831
    + 138318954125048704418830512450804931
    step 72: 277727008340087518826671033910618762
    + 267816019330176628815780043800727772
    step 73: 545543027670264147642451077711346534
    + 435643117770154246741462076720345545
    step 74: 981186145440418394383913154431692079
    + 970296134451319383493814044541681189
    step 75: 1951482279891737777877727198973373268
    + 8623733798917277787777371989722841591
    step 76: 10575216078809015565655099188696214859
    + 95841269688199055656551090887061257501
    step 77: 106416485767008071222206190075757472360
    + 063274757570091602222170800767584614601
    step 78: 169691243337099673444376990843342086961
    + 169680243348099673444376990733342196961
    step 79: 339371486685199346888753981576684283922
    + 229382486675189357888643991586684173933
    step 80: 568753973360388704777397973163368457855
    + 558754863361379793777407883063379357865
    step 81: 1127508836721768498554805856226747815720
    + 0275187476226585084558948671276388057211
    step 82: 1402696312948353583113754527503135872931
    + 1392785313057254573113853538492136962041
    step 83: 2795481626005608156227608065995272834972
    + 2794382725995608067226518065006261845972
    step 84: 5589864352001216223454126131001534680944
    + 4490864351001316214543226121002534689855
    step 85: 10080728703002532437997352252004069370799
    + 99707396040025225379973423520030782708001
    step 86: 109788124743027757817970775772034852078800
    + 008870258430277577079718757720347421887901
    step 87: 118658383173305334897689533492382273966701
    + 107669372283294335986798433503371383856811
    step 88: 226327755456599670884487966995753657823512
    + 215328756357599669784488076995654557723622
    step 89: 441656511814199340668976043991408215547134
    + 431745512804199340679866043991418115656144
    step 90: 873402024618398681348842087982826331203278
    + 872302133628289780248843186893816420204378
    step 91: 1745704158246688461597685274876642751407656
    + 6567041572466784725867951648866428514075471
    step 92: 8312745730713473187465636923743071265483127
    + 7213845621703473296365647813743170375472138
    step 93: 15526591352416946483831284737486241640955265
    + 56255904614268473748213838464961425319562551
    step 94: 71782495966685420232045123202447666960517816
    + 61871506966674420232154023202458666959428717
    step 95: 133654002933359840464199146404906333919946533
    + 335649919333609404641991464048953339200456331
    step 96: 469303922266969245106190610453859673120402864
    + 468204021376958354016091601542969662229303964
    step 97: 937507943643927599122282211996829335349706828
    + 828607943533928699112282221995729346349705739
    step 98: 1766115887177856298234564433992558681699412567
    + 7652149961868552993344654328926587717885116671
    step 99: 9418265849046409291579218762919146399584529238
    + 8329254859936419192678129751929046409485628149
    step 100: 17747520708982828484257348514848192809070157387
    + 78375107090829184841584375248482828980702574771
    step 101: 96122627799812013325841723763331021789772732158
    + 85123727798712013336732714852331021899772622169
    step 102: 181246355598524026662574438615662043689545354327
    + 723453545986340266516834475266620425895553642181
    step 103: 904699901584864293179408913882282469585098996508
    + 805699890585964282288319804971392468485109996409
    step 104: 1710399792170828575467728718853674938070208992917
    + 7192998020708394763588178277645758280712979930171
    step 105: 8903397812879223339055906996499433218783188923088
    + 8803298813878123349946996095509333229782187933098
    step 106: 17706696626757346689002903092008766448565376856186
    + 68165867356584466780029030920098664375762669660771
    step 107: 85872563983341813469031934012107430824328046516957
    + 75961564082342803470121043913096431814338936527858
    step 108: 161834128065684616939152977925203862638666983044815
    + 518440389666836268302529779251939616486560821438161
    step 109: 680274517732520885241682757177143479125227804482976
    + 679284408722521974341771757286142588025237715472086
    step 110: 1359558926455042859583454514463286067150465519955062
    + 2605599155640517606823644154543859582405546298559531
    step 111: 3965158082095560466407098669007145649556011818514593
    + 3954158181106559465417009668907046640655902808515693
    step 112: 7919316263202119931824108337914192290211914627030286
    + 6820307264191120922914197338014281399112023626139197
    step 113: 14739623527393240854738305675928473689323938253169483
    + 38496135283932398637482957650383745804239372532693741
    step 114: 53235758811325639492221263326312219493563310785863224
    + 42236858701336539491221362336212229493652311885753235
    step 115: 95472617512662178983442625662524448987215622671616459
    + 95461617622651278984442526652624438987126621571627459
    step 116: 190934235135313457967885152315148887974342244243243918
    + 819342342442243479788841513251588769754313531532439091
    step 117: 1010276577577556937756726665566737657728655775775683009
    + 9003865775775568277567376655666276577396557757756720101
    step 118: 10014142353353125215324103321233014235125213533532403110
    + 01130423533531252153241033212330142351252135335324141001
    step 119: 11144565886884377368565136533563156586377348868856544111
    18317699990 takes 119 iterations / steps to resolve into a 56 digit palindrome.

    REVERSAL-ADDITION PALINDROME RECORDS

    Most Delayed Palindromic Number for each digit length
    (Only iteration counts for which no smaller records exist are considered. My program records only the smallest number that resolves for each distinct iteration count. For example, there are 18-digit numbers that resolve in 232 iterations, higher than the 228 iteration record shown for 18-digit numbers, but they were not recorded, as a smaller [17-digit] number already holds the record for 232 iterations.)

    DigitsNumberResult
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    89
    187
    1,297
    10,911
    150,296
    9,008,299
    10,309,988
    140,669,390
    1,005,499,526
    10,087,799,570
    100,001,987,765
    1,600,005,969,190
    14,104,229,999,995
    100,120,849,299,260
    1,030,020,097,997,900
    10,442,000,392,399,960
    170,500,000,303,619,996
    1,186,060,307,891,929,990
    solves in 24 iterations.
    solves in 23 iterations.
    solves in 21 iterations.
    solves in 55 iterations.
    solves in 64 iterations.
    solves in 96 iterations.
    solves in 95 iterations.
    solves in 98 iterations.
    solves in 109 iterations.
    solves in 149 iterations.
    solves in 143 iterations.
    solves in 188 iterations.
    solves in 182 iterations.
    solves in 201 iterations.
    solves in 197 iterations.
    solves in 236 iterations.
    solves in 228 iterations.
    solves in 261 iterations - World Record!
    [View all records]

    This reverse and add program was created by Jason Doucette.
    Please visit my Palindromes and World Records page.
    You have permission to use the data from this webpage (with due credit).
    A link to my website is much appreciated. Thank you.

    (This program has been run 2,061,844 times since Saturday, March 9th, 2002.)
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