山水娱乐

  • REVERSAL-ADDITION PALINDROME TEST ON 1090001921

    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 1090001921:
    1090001921
    + 1291000901
    step 1: 2381002822
    + 2282001832
    step 2: 4663004654
    + 4564003664
    step 3: 9227008318
    + 8138007229
    step 4: 17365015547
    + 74551056371
    step 5: 91916071918
    + 81917061919
    step 6: 173833133837
    + 738331338371
    step 7: 912164472208
    + 802274461219
    step 8: 1714438933427
    + 7243398344171
    step 9: 8957837277598
    + 8957727387598
    step 10: 17915564665196
    + 69156646551971
    step 11: 87072211217167
    + 76171211227078
    step 12: 163243422444245
    + 542444224342361
    step 13: 705687646786606
    + 606687646786507
    step 14: 1312375293573113
    + 3113753925732131
    step 15: 4426129219305244
    + 4425039129216244
    step 16: 8851168348521488
    + 8841258438611588
    step 17: 17692426787133076
    + 67033178762429671
    step 18: 84725605549562747
    + 74726594550652748
    step 19: 159452200100215495
    + 594512001002254951
    step 20: 753964201102470446
    + 644074201102469357
    step 21: 1398038402204939803
    + 3089394022048308931
    step 22: 4487432424253248734
    + 4378423524242347844
    step 23: 8865855948495596578
    + 8756955948495585688
    step 24: 17622811896991182266
    + 66228119969811822671
    step 25: 83850931866803004937
    + 73940030866813905838
    step 26: 157790962733616910775
    + 577019616337269097751
    step 27: 734810579070886008526
    + 625800688070975018437
    step 28: 1360611267141861026963
    + 3696201681417621160631
    step 29: 5056812948559482187594
    + 4957812849558492186505
    step 30: 10014625798117974374099
    + 99047347971189752641001
    step 31: 109061973769307727015100
    + 001510727703967379160901
    step 32: 110572701473275106176001
    + 100671601572374107275011
    step 33: 211244303045649213451012
    + 210154312946540303442112
    step 34: 421398615992189516893124
    + 421398615981299516893124
    step 35: 842797231973489033786248
    + 842687330984379132797248
    step 36: 1685484562957868166583496
    + 6943856618687592654845861
    step 37: 8629341181645460821429357
    + 7539241280645461811439268
    step 38: 16168582462290922632868625
    + 52686823622909226428586161
    step 39: 68855406085200149061454786
    + 68745416094100258060455886
    step 40: 137600822179300407121910672
    + 276019121704003971228006731
    step 41: 413619943883304378349917403
    + 304719943873403388349916314
    step 42: 718339887756707766699833717
    + 717338996667707657788933817
    step 43: 1435678884424415424488767534
    + 4357678844245144244888765341
    step 44: 5793357728669559669377532875
    + 5782357739669559668277533975
    step 45: 11575715468339119337655066850
    + 05866055673391193386451757511
    step 46: 17441771141730312724106824361
    + 16342860142721303714117714471
    step 47: 33784631284451616438224538832
    + 23883542283461615448213648733
    step 48: 57668173567913231886438187565
    + 56578183468813231976537186675
    step 49: 114246357036726463862975374240
    + 042473579268364627630753642411
    step 50: 156719936305091091493729016651
    + 156610927394190190503639917651
    step 51: 313330863699281281997368934302
    + 203439863799182182996368033313
    step 52: 516770727498463464993736967615
    + 516769637399464364894727077615
    step 53: 1033540364897927829888464045230
    + 0325404648889287297984630453301
    step 54: 1358945013787215127873094498531
    + 1358944903787215127873105498531
    step 55: 2717889917574430255746199997062
    + 2607999916475520344757199887172
    step 56: 5325889834049950600503399884234
    + 4324889933050060599404389885235
    step 57: 9650779767100011199907789769469
    + 9649679877099911100017679770569
    step 58: 19300459644199922299925469540038
    + 83004596452999222999144695400391
    step 59: 102305056097199145299070164940429
    + 924049461070992541991790650503201
    step 60: 1026354517168191687290860815443630
    + 0363445180680927861918617154536201
    step 61: 1389799697849119549209477969979831
    + 1389799697749029459119487969979831
    step 62: 2779599395598149008328965939959662
    + 2669599395698238009418955939959772
    step 63: 5449198791296387017747921879919434
    + 4349199781297477107836921978919445
    step 64: 9798398572593864125584843858838879
    + 9788388583484855214683952758938979
    step 65: 19586787156078719340268796617777858
    + 85877771669786204391787065178768591
    step 66: 105464558825864923732055861796546449
    + 944645697168550237329468528855464501
    step 67: 1050110255994415161061524390652010950
    + 0590102560934251601615144995520110501
    step 68: 1640212816928666762676669386172121451
    + 1541212716839666762676668296182120461
    step 69: 3181425533768333525353337682354241912
    + 2191424532867333535253338673355241813
    step 70: 5372850066635667060606676355709483725
    + 5273849075536766060607665366600582735
    step 71: 10646699142172433121214341722310066460
    + 06466001322714341212133427124199664601
    step 72: 17112700464886774333347768846509731061
    + 16013790564886774333347768846400721171
    step 73: 33126491029773548666695537692910452232
    + 23225401929673559666684537792019462133
    step 74: 56351892959447108333380075484929914365
    + 56341992948457008333380174495929815365
    step 75: 112693885907904116666760249980859729730
    + 037927958089942067666611409709588396211
    step 76: 150621843997846184333371659690448125941
    + 149521844096956173333481648799348126051
    step 77: 300143688094802357666853308489796251992
    + 299152697984803358666753208490886341003
    step 78: 599296386079605716333606516980682592995
    + 599295286089615606333617506970683692995
    step 79: 1198591672169221322667224023951366285990
    + 0995826631593204227662231229612761958911
    step 80: 2194418303762425550329455253564128244901
    + 1094428214653525549230555242673038144912
    step 81: 3288846518415951099560010496237166389813
    + 3189836617326940100659901595148156488823
    step 82: 6478683135742891200219912091385322878636
    + 6368782235831902199120021982475313868746
    step 83: 12847465371574793399339934073860636747382
    + 28374763606837043993399339747517356474821
    step 84: 41222228978411837392739273821377993222203
    + 30222239977312837293729373811487982222214
    step 85: 71444468955724674686468647632865975444417
    + 71444457956823674686468647642755986444417
    step 86: 142888926912548349372937295275621961888834
    + 438888169126572592739273943845219629888241
    step 87: 581777096039120942112211239120841591777075
    + 570777195148021932112211249021930690777185
    step 88: 1152554291187142874224422488142772282554260
    + 0624552822772418842244224782417811924552511
    step 89: 1777107113959561716468647270560584207106771
    + 1776017024850650727468646171659593117017771
    step 90: 3553124138810212443937293442220177324124542
    + 2454214237710222443927393442120188314213553
    step 91: 6007338376520434887864686884340365638338095
    + 5908338365630434886864687884340256738337006
    step 92: 11915676742150869774729374768680622376675101
    + 10157667322608686747392747796805124767651911
    step 93: 22073344064759556522122122565485747144327012
    + 21072344174758456522122122565595746044337022
    step 94: 43145688239518013044244245131081493188664034
    + 43046688139418013154244244031081593288654134
    step 95: 86192376378936026198488489162163086477318168
    + 86181377468036126198488489162063987367329168
    step 96: 172373753846972152396976978324227073844647336
    + 633746448370722423879679693251279648357373271
    step 97: 806120202217694576276656671575506722202020607
    + 706020202227605575176656672675496712202021608
    step 98: 1512140404445300151453313344251003434404042215
    + 5122404044343001524433133541510035444040412151
    step 99: 6634544448788301675886446885761038878444454366
    1090001921 takes 99 iterations / steps to resolve into a 46 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 2,061,694 times since Saturday, March 9th, 2002.)
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