山水娱乐

  • REVERSAL-ADDITION PALINDROME TEST ON 110909992

    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 110909992:
    110909992
    + 299909011
    step 1: 410819003
    + 300918014
    step 2: 711737017
    + 710737117
    step 3: 1422474134
    + 4314742241
    step 4: 5737216375
    + 5736127375
    step 5: 11473343750
    + 05734337411
    step 6: 17207681161
    + 16118670271
    step 7: 33326351432
    + 23415362333
    step 8: 56741713765
    + 56731714765
    step 9: 113473428530
    + 035824374311
    step 10: 149297802841
    + 148208792941
    step 11: 297506595782
    + 287595605792
    step 12: 585102201574
    + 475102201585
    step 13: 1060204403159
    + 9513044020601
    step 14: 10573248423760
    + 06732484237501
    step 15: 17305732661261
    + 16216623750371
    step 16: 33522356411632
    + 23611465322533
    step 17: 57133821734165
    + 56143712833175
    step 18: 113277534567340
    + 043765435772311
    step 19: 157042970339651
    + 156933079240751
    step 20: 313976049580402
    + 204085940679313
    step 21: 518061990259715
    + 517952099160815
    step 22: 1036014089420530
    + 0350249804106301
    step 23: 1386263893526831
    + 1386253983626831
    step 24: 2772517877153662
    + 2663517787152772
    step 25: 5436035664306434
    + 4346034665306345
    step 26: 9782070329612779
    + 9772169230702879
    step 27: 19554239560315658
    + 85651306593245591
    step 28: 105205546153561249
    + 942165351645502501
    step 29: 1047370897799063750
    + 0573609977980737401
    step 30: 1620980875779801151
    + 1511089775780890261
    step 31: 3132070651560691412
    + 2141960651560702313
    step 32: 5274031303121393725
    + 5273931213031304725
    step 33: 10547962516152698450
    + 05489625161526974501
    step 34: 16037587677679672951
    + 15927697677678573061
    step 35: 31965285355358246012
    + 21064285355358256913
    step 36: 53029570710716502925
    + 52920561701707592035
    step 37: 105950132412424094960
    + 069490424214231059501
    step 38: 175440556626655154461
    + 164451556626655044571
    step 39: 339892113253310199032
    + 230991013352311298933
    step 40: 570883126605621497965
    + 569794126506621388075
    step 41: 1140677253112242886040
    + 0406882422113527760411
    step 42: 1547559675225770646451
    + 1546460775225769557451
    step 43: 3094020450451540203902
    + 2093020451540540204903
    step 44: 5187040901992080408805
    + 5088040802991090407815
    step 45: 10275081704983170816620
    + 02661807138940718057201
    step 46: 12936888843923888873821
    + 12837888832934888863921
    step 47: 25774777676858777737742
    + 24773777785867677747752
    step 48: 50548555462726455485494
    + 49458455462726455584505
    step 49: 100007010925452911069999
    + 999960119254529010700001
    step 50: 1099967130179981921770000
    + 0000771291899710317699901
    step 51: 1100738422079692239469901
    + 1099649322969702248370011
    step 52: 2200387745049394487839912
    + 2199387844939405477830022
    step 53: 4399775589988799965669934
    + 4399665699978899855779934
    step 54: 8799441289967699821449868
    + 8689441289967699821449978
    step 55: 17488882579935399642899846
    + 64899824699353997528888471
    step 56: 82388707279289397171788317
    + 71388717179398297270788328
    step 57: 153777424458687694442576645
    + 546675244496786854424777351
    step 58: 700452668955474548867353996
    + 699353768845474559866254007
    step 59: 1399806437800949108733608003
    + 3008063378019490087346089931
    step 60: 4407869815820439196079697934
    + 4397969706919340285189687044
    step 61: 8805839522739779481269384978
    + 8794839621849779372259385088
    step 62: 17600679144589558853528770066
    + 66007782535885598544197600671
    step 63: 83608461680475157397726370737
    + 73707362779375157408616480638
    step 64: 157315824459850314806342851375
    + 573158243608413058954428513751
    step 65: 730474068068263373760771365126
    + 621563177067373362860860474037
    step 66: 1352037245135636736621631839163
    + 3619381361266376365315427302531
    step 67: 4971418606402013101937059141694
    + 4961419507391013102046068141794
    step 68: 9932838113793026203983127283488
    + 8843827213893026203973118382399
    step 69: 18776665327686052407956245665887
    + 78856654265970425068672356667781
    step 70: 97633319593656477476628602333668
    + 86633320682667477465639591333679
    step 71: 184266640276323954942268193667347
    + 743766391862249459323672046662481
    step 72: 928033032138573414265940240329828
    + 828923042049562414375831230330829
    step 73: 1756956074188135828641771470660657
    + 7560660741771468285318814706596571
    step 74: 9317616815959604113960586177257228
    + 8227527716850693114069595186167139
    step 75: 17545144532810297228030181363424367
    + 76342436318103082279201823544154571
    step 76: 93887580850913379507232004907578938
    + 83987570940023270597331905808578839
    step 77: 177875151790936650104563910716157777
    + 777751617019365401056639097151578771
    step 78: 955626768810302051161203007867736548
    + 845637768700302161150203018867626559
    step 79: 1801264537510604212311406026735363107
    + 7013635376206041132124060157354621081
    step 80: 8814899913716645344435466184089984188
    + 8814899804816645344435466173199984188
    step 81: 17629799718533290688870932357289968376
    + 67386998275323907888609233581799792671
    step 82: 85016797993857198577480165939089761047
    + 74016798093956108477589175839979761058
    step 83: 159033596087813307055069341779069522105
    + 501225960977143960550703318780695330951
    step 84: 660259557064957267605772660559764853056
    + 650358467955066277506762759460755952066
    step 85: 1310618025020023545112535420020520805122
    + 2215080250200245352115453200205208160131
    step 86: 3525698275220268897227988620225728965253
    110909992 takes 86 iterations / steps to resolve into a 40 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 2,061,817 times since Saturday, March 9th, 2002.)
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