山水娱乐

  • REVERSAL-ADDITION PALINDROME TEST ON 10309988

    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 10309988:
    10309988
    + 88990301
    step 1: 99300289
    + 98200399
    step 2: 197500688
    + 886005791
    step 3: 1083506479
    + 9746053801
    step 4: 10829560280
    + 08206592801
    step 5: 19036153081
    + 18035163091
    step 6: 37071316172
    + 27161317073
    step 7: 64232633245
    + 54233623246
    step 8: 118466256491
    + 194652664811
    step 9: 313118921302
    + 203129811313
    step 10: 516248732615
    + 516237842615
    step 11: 1032486575230
    + 0325756842301
    step 12: 1358243417531
    + 1357143428531
    step 13: 2715386846062
    + 2606486835172
    step 14: 5321873681234
    + 4321863781235
    step 15: 9643737462469
    + 9642647373469
    step 16: 19286384835938
    + 83953848368291
    step 17: 103240233204229
    + 922402332042301
    step 18: 1025642565246530
    + 0356425652465201
    step 19: 1382068217711731
    + 1371177128602831
    step 20: 2753245346314562
    + 2654136435423572
    step 21: 5407381781738134
    + 4318371871837045
    step 22: 9725753653575179
    + 9715753563575279
    step 23: 19441507217150458
    + 85405171270514491
    step 24: 104846678487664949
    + 949466784876648401
    step 25: 1054313463364313350
    + 0533134633643134501
    step 26: 1587448097007447851
    + 1587447007908447851
    step 27: 3174895104915895702
    + 2075985194015984713
    step 28: 5250880298931880415
    + 5140881398920880525
    step 29: 10391761697852760940
    + 04906725879616719301
    step 30: 15298487577469480241
    + 14208496477578489251
    step 31: 29506984055047969492
    + 29496974055048960592
    step 32: 59003958110096930084
    + 48003969001185930095
    step 33: 107007927111282860179
    + 971068282111729700701
    step 34: 1078076209223012560880
    + 0880652103229026708701
    step 35: 1958728312452039269581
    + 1859629302542138278591
    step 36: 3818357614994177548172
    + 2718457714994167538183
    step 37: 6536815329988345086355
    + 5536805438899235186356
    step 38: 12073620768887580272711
    + 11727208578886702637021
    step 39: 23800829347774282909732
    + 23790928247774392800832
    step 40: 47591757595548675710564
    + 46501757684559575719574
    step 41: 94093515280108251430138
    + 83103415280108251539049
    step 42: 177196930560216502969187
    + 781969205612065039691771
    step 43: 959166136172281542660958
    + 859066245182271631661959
    step 44: 1818232381354553174322917
    + 7192234713554531832328181
    step 45: 9010467094909085006651098
    + 8901566005809094907640109
    step 46: 17912033100718179914291207
    + 70219241997181700133021971
    step 47: 88131275097899880047313178
    + 87131374008899879057213188
    step 48: 175262649106799759104526366
    + 663625401957997601946262571
    step 49: 838888051064797361050788937
    + 739887050163797460150888838
    step 50: 1578775101228594821201677775
    + 5777761021284958221015778751
    step 51: 7356536122513553042217456526
    + 6256547122403553152216356537
    step 52: 13613083244917106194433813063
    + 36031833449160171944238031631
    step 53: 49644916694077278138671844694
    + 49644817683187277049661944694
    step 54: 99289734377264555188333789388
    + 88398733388155546277343798299
    step 55: 187688467765420101465677587687
    + 786785776564101024567764886781
    step 56: 974474244329521126033442474468
    + 864474244330621125923442474479
    step 57: 1838948488660142251956884948947
    + 7498494886591522410668848498381
    step 58: 9337443375251664662625733447328
    + 8237443375262664661525733447339
    step 59: 17574886750514329324151466894667
    + 76649866415142392341505768847571
    step 60: 94224753165656721665657235742238
    + 83224753275656612765656135742249
    step 61: 177449506441313334431313371484487
    + 784484173313134433313144605944771
    step 62: 961933679754447767744457977429258
    + 852924779754447767744457976339169
    step 63: 1814858459508895535488915953768427
    + 7248673595198845355988059548584181
    step 64: 9063532054707740891476975502352608
    + 8062532055796741980477074502353609
    step 65: 17126064110504482871954050004706217
    + 71260740005045917828440501146062171
    step 66: 88386804115550400700394551150768388
    + 88386705115549300700405551140868388
    step 67: 176773509231099701400800102291636776
    + 677636192201008004107990132905377671
    step 68: 854409701432107705508790235197014447
    + 744410791532097805507701234107904458
    step 69: 1598820492964205511016491469304918905
    + 5098194039641946101155024692940288951
    step 70: 6697014532606151612171516162245207856
    + 6587025422616151712161516062354107966
    step 71: 13284039955222303324333032224599315822
    + 22851399542223033342330322255993048231
    step 72: 36135439497445336666663354480592364053
    + 35046329508445336666663354479493453163
    step 73: 71181769005890673333326708960085817216
    + 61271858006980762333337609850096718117
    step 74: 132453627012871435666664318810182535333
    + 333535281018813466666534178210726354231
    step 75: 465988908031684902333198497020908889564
    + 465988809020794891333209486130809889564
    step 76: 931977717052479793666407983151718779128
    + 821977817151389704666397974250717779139
    step 77: 1753955534203869498332805957402436558267
    + 7628556342047595082338949683024355593571
    step 78: 9382511876251464580671755640426792151838
    + 8381512976240465571760854641526781152839
    step 79: 17764024852491930152432610281953573304677
    + 77640337535918201623425103919425842046771
    step 80: 95404362388410131775857714201379415351448
    + 84415351497310241775857713101488326340459
    step 81: 179819713885720373551715427302867741691907
    + 709196147768203724517155373027588317918971
    step 82: 889015861653924098068870800330456059610878
    + 878016950654033008078860890429356168510988
    step 83: 1767032812307957106147731690759812228121866
    + 6681218222189570961377416017597032182307671
    step 84: 8448251034497528067525147708356844410429537
    + 7359240144486538077415257608257944301528448
    step 85: 15807491178984066144940405316614788711957985
    + 58975911788741661350404944166048987119470851
    step 86: 74783402967725727495345349482663775831428836
    + 63882413857736628494354359472752776920438747
    step 87: 138665816825462355989699708955416552751867583
    + 385768157255614559807996989553264528618566831
    step 88: 524433974081076915797696698508681081370434414
    + 414434073180186805896696797519670180479334425
    step 89: 938868047261263721694393496028351261849768839
    + 938867948162153820694393496127362162740868839
    step 90: 1877735995423417542388786992155713424590637678
    + 8767360954243175512996878832457143245995377781
    step 91: 10645096949666593055385665824612856670586015459
    + 95451068507665821642856658355039566694969054601
    step 92: 106096165457332414698242324179652423365555070060
    + 060070555563324256971423242896414233754561690601
    step 93: 166166721020656671669665567076066657120116760661
    + 166067611021756660670765566966176656020127661661
    step 94: 332234332042413332340431134042243313140244422322
    + 223224442041313342240431134043233314240233432233
    step 95: 555458774083726674580862268085476627380477854555
    10309988 takes 95 iterations / steps to resolve into a 48 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 2,061,815 times since Saturday, March 9th, 2002.)
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