Jason Doucette / Xona Games
 
hometown – Yarmouth, NS, Canada
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196 Palindrome Quest

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196 Iterated 200 Times via Reversal-Addition

This page shows 196 Palindrome Quest (a.k.a. 196 Algorithm, or 196 Problem) iterated 200 times via reverse-addition. This is a sub page of my World Records (Palindrome) Page, which fully explains my involvement in The 196 Palindrome Quest / Algorithm / Problem, among others. It also shows another one of my palindrome records, The Most Delayed Palindromic Number. Please visit these pages, if you wish to learn more. The page you are currently reading was meant only to show the initial 200 iterations of the 196 Palindrome Quest, and to give an idea of the magnitude of calculations required to take this quest into millions of digits.

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The 196 Algorithm:
  1. Take a number.
  2. Reverse its digits, and add.
  3. If the result is not a palindrome (i.e. reads forwards as it does backwards), then repeat.
Sounds easy? Let's try it for 196:

196
+ 691
step 1: 887
+ 788
step 2: 1675
+ 5761
step 3: 7436
+ 6347
step 4: 13783
+ 38731
step 5: 52514
+ 41525
step 6: 94039
+ 93049
step 7: 187088
+ 880781
step 8: 1067869
+ 9687601
step 9: 10755470
+ 07455701
step 10: 18211171
+ 17111281
step 11: 35322452
+ 25422353
step 12: 60744805
+ 50844706
step 13: 111589511
+ 115985111
step 14: 227574622
+ 226475722
step 15: 454050344
+ 443050454
step 16: 897100798
+ 897001798
step 17: 1794102596
+ 6952014971
step 18: 8746117567
+ 7657116478
step 19: 16403234045
+ 54043230461
step 20: 70446464506
+ 60546464407
step 21: 130992928913
+ 319829299031
step 22: 450822227944
+ 449722228054
step 23: 900544455998
+ 899554445009
step 24: 1800098901007
+ 7001098900081
step 25: 8801197801088
+ 8801087911088
step 26: 17602285712176
+ 67121758220671
step 27: 84724043932847
+ 74823934042748
step 28: 159547977975595
+ 595579779745951
step 29: 755127757721546
+ 645127757721557
step 30: 1400255515443103
+ 3013445155520041
step 31: 4413700670963144
+ 4413690760073144
step 32: 8827391431036288
+ 8826301341937288
step 33: 17653692772973576
+ 67537927729635671
step 34: 85191620502609247
+ 74290620502619158
step 35: 159482241005228405
+ 504822500142284951
step 36: 664304741147513356
+ 653315741147403466
step 37: 1317620482294916822
+ 2286194922840267131
step 38: 3603815405135183953
+ 3593815315045183063
step 39: 7197630720180367016
+ 6107630810270367917
step 40: 13305261530450734933
+ 33943705403516250331
step 41: 47248966933966985264
+ 46258966933966984274
step 42: 93507933867933969538
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step 43: 177104867844767940077
+ 770049767448768401771
step 44: 947154635293536341848
+ 848143635392536451749
step 45: 1795298270686072793597
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step 46: 9749270977546801719568
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step 47: 18408442064004592449047
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step 48: 92502871604050616929528
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step 49: 175095833209091234750057
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step 51: 1751196640799987135692157
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step 53: 17537224026299926194218357
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step 55: 175837936477498486373973857
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step 56: 934217310162393261013712428
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step 57: 1758434620324786522027424867
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step 58: 9442681822581660752291773438
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step 59: 17786453745152322604573635887
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step 60: 96640091285774647759309104658
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step 61: 182280281681549295517528109327
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step 62: 906182107397142240703710191608
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step 63: 1712373124704184482497411473217
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step 64: 8836114272647029296571625205388
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step 65: 17671139534403958504034349321776
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step 66: 85383533877444544434477942439447
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step 67: 159876958854887988978955775977805
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step 68: 668656536414767878767414635656756
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step 69: 1326313072829535757534829271313622
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step 70: 3589444802113893332894111974449853
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step 71: 7178889593228875666877224058899706
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step 72: 13258878097456662332665448018788423
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step 73: 45747659181913285659330927106673654
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step 74: 91385319354816681317562845302348408
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step 75: 171869639709643252636224690693706727
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step 76: 899477035806065888888571598630674898
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step 78: 8787394689723559555548553279865047867
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step 81: 131467596250025596244378551114170566423
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step 82: 456132667661181469687074071166866330554
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step 84: 1713431572655604770948087405557266223327
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step 85: 8946658200210652579438861471120017566498
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step 86: 17893315300422394267788614031240046132996
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step 87: 87816479304635435956564863353640397472867
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step 88: 164643958609270772803130816807280794934745
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step 89: 712083455691979390834439093880187654281206
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step 90: 1314265912473067781768877187859384208661423
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step 91: 4555933937312655599557549065463126404285554
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step 92: 9111757983526301209015109021025263797681108
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step 93: 17123625957151502418030218042061517695252227
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step 94: 89348885628667526499233299462576693647884398
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step 95: 178697760268335052998466598925153376306768796
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step 96: 876565363941686582894131498175687238374565667
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step 97: 1643130837774473154788262996461373387738131345
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step 98: 7074449215608204801780891870975118165118444806
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step 99: 13158897331226320592562872742059146230247889513
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step 100: 44757771534490515617290699271561508443627774644
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step 101: 89405544168971032134590308543213017887145550388
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step 102: 177711098347842063369170618086336035873290100886
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step 103: 865712190726372697049986690049696284617180218657
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step 104: 1622524272442855393990083379990492558244271436225
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step 105: 6848865996871408334989817180984428140686995688486
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step 106: 13697731993731826579880634370878766182473991376972
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step 107: 41665051931159993367687977979776328996213905156603
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step 108: 72330202862429975735485955958452668991327820213217
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step 109: 143561405734749962360971911916906426983754640416544
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step 110: 589175452192139586970591031095969696931192144581885
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step 111: 1177360893483279283940181161291049382862483399153870
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step 112: 1960880827325962123342102773101543212586327379791581
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step 113: 3812860564562814246793116545113976425281564660672272
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step 114: 6535621229214639493586232001227952849464219311354455
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step 115: 12080152368339288976183453003554806798828348532619811
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step 116: 23971775952722178737028983038992974787121734857727832
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step 117: 47844551796434357484958966077975048574244460815445764
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step 118: 94599003602878605069016943144960997049587930530990638
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step 119: 178208907106857199148923887279922093100275751161090187
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step 120: 959299068264429200539153860068251935092034352870893058
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step 121: 1809697146517859491078306720136603870094958815731886017
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step 122: 8916578521706454391861373030412642572044545972149855098
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step 123: 17822167934501908794613835170716374253979092043408611296
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step 124: 87033848368531006729861196877870205903760002587384734167
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step 125: 163177596747051013460811404755739322796520016173769567245
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step 126: 705943564118661039158035342313143440860830166921465338606
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step 127: 1312777128248322077226079683626386971712760333732930688113
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step 128: 4431637520621652749397876519890256677940462572161148460244
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step 129: 8852285932234405389895643040879413465879935133421405821588
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step 130: 17703570973477720789681286190659816931869770177743801644176
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step 131: 84848181808254828586495248086268985150568472955181709174947
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step 132: 159795372526410756073000407072537069410037055800462527359795
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step 133: 757749097790419306803015367807807773410407712815087800957746
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step 134: 1405508106570937524507029745516516536920716316729185591905503
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step 135: 4460600062390213660677326101672672016127770574119941609960544
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step 136: 8911299123889328411454542204435433032365531237240874210021188
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step 137: 17722499248669755732810174507780777054820072385480757429942386
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step 138: 86047491724378214059813019585489547601921896141277441729365157
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step 139: 161203884438855428229625930260088006192953791182564784448839225
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step 155: 84720287247484417510300098966384988839466978000410482437485169301747
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step 165: 1361223042963149970271554446149181758868072041635555062080050379140233063
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step 169: 97449922115672501335590732544163267577467336034533708553310627751121905468
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step 170: 183900834231445102671171466087226644055043572179057418106621155402244899947
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step 171: 933899276435996229272986217058501984605490194959721589282822699534682909328
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step 172: 1757808562871992457555971345017993079111979300810434278555745399069355907667
+ 7667095539609935475558724340180039791119703997105431795557542991782658087571
step 173: 9424904102481927933114695685198032870231683297915866074113288390852013995238
+ 8325993102580938823114706685197923861320782308915865964113397291842014094249
step 174: 17750897205062866756229402370395956731552465606831732038226685682694028089487
+ 78498082049628658662283023713860656425513765959307320492265766826050279805771
step 175: 96248979254691525418512426084256613157066231566139052530492452508744307895258
+ 85259870344780525429403525093166513266075131665248062421581452519645297984269
step 176: 181508849599472050847915951177423126423141363231387114952073905028389605879527
+ 725978506983820509370259411783132363141324621324771159519748050274995948805181
step 177: 907487356583292560218175362960555489564465984556158274471821955303385554684708
+ 807486455583303559128174472851655489564465984555069263571812065292385653784709
step 178: 1714973812166596119346349835812210979128931969111227538043634020595771208469417
+ 7149648021775950204363408357221119691398219790122185389436439116956612183794171
step 179: 8864621833942546323709758193033330670527151759233412927480073137552383392263588
+ 8853622933832557313700847292143329571517250760333303918579073236452493381264688
step 180: 17718244767775103637410605485176660242044402519566716846059146374004876773528276
+ 67282537767840047364195064861766591520444024206667158450601473630157776744281771
step 181: 85000782535615151001605670346943251762488426726233875296660620004162653517810047
+ 74001871535626140002606669257833262762488426715234964307650610015151653528700058
step 182: 159002654071241291004212339604776514524976853441468839604311230019314307046510105
+ 501015640703413910032113406938864144358679425415677406933212400192142170456200951
step 183: 660018294774655201036325746543640658883656278857146246537523630211456477502711056
+ 650117205774654112036325735642641758872656388856046345647523630102556477492810066
step 184: 1310135500549309313072651482186282417756312667713192592185047260314012954995521122
+ 2211255994592104130627405812952913177662136577142826812841562703139039450055310131
step 185: 3521391495141413443700057295139195595418449244856019405026609963453052405050831253
+ 3521380505042503543699066205049106584429448145955919315927500073443141415941931253
step 186: 7042772000183916987399123500188302179847897390811938720954110036896193820992762506
+ 6052672990283916986300114590278391180937987489712038810053219937896193810002772407
step 187: 13095444990467833973699238090466693360785884880523977531007329974792387630995534913
+ 31943559903678329747992370013577932508848858706339666409083299637933876409944459031
step 188: 45039004894146163721691608104044625869634743586863643940090629612726264040939993944
+ 44939993904046262721692609004934636868534743696852644040180619612736164149840093054
step 189: 89978998798192426443384217108979262738169487283716287980271249225462428190780086998
+ 89968008709182426452294217208978261738278496183726297980171248334462429189789987998
step 190: 179947007507374852895678434317957524476447983467442585960442497559924857380570074996
+ 699470075083758429955794244069585244764389744674425759713434876598258473705700749971
step 191: 879417082591133282851472678387542769240837728141868345673877374158183331086270824967
+ 769428072680133381851473778376543868141827738042967245783876274158282331195280714978
step 192: 1648845155271266664702946456764086637382665466184835591457753648316465662281551539945
+ 5499351551822665646138463577541955384816645662837366804676546492074666621725515488461
step 193: 7148196707093932310841410034306042022199311129022202396134300140391132284007067028406
+ 6048207607004822311930410034316932022209211139912202406034300141480132393907076918417
step 194: 13196404314098754622771820068622974044408522268934404802168600281871264677914143946823
+ 32864934141977646217818200686120840443986222580444047922686002817722645789041340469131
step 195: 46061338456076400840590020754743814488394744849378452724854603099593910466955484415954
+ 45951448455966401939599030645842725487394844749388441834745702009504800467065483316064
step 196: 92012786912042802780189051400586539975789589598766894559600305109098710934020967732018
+ 81023776902043901789090150300695549866789598598757993568500415098108720824021968721029
step 197: 173036563814086704569279201701282089842579188197524888128100720207207431758042936453047
+ 740354639240857134702702027001821888425791881975248980282107102972965407680418365630371
step 198: 913391203054943839271981228703103978268371070172773868410207823180172839438461302083418
+ 814380203164834938271081328702014868377271070173862879301307822189172938349450302193319
step 199: 1727771406219778777543062557405118846645642140346636747711515645369345777787911604276737
+ 7376724061197877775439635465151177476366430412465466488115047552603457778779126041777271
step 200: 9104495467417656552982698022556296323012072552812103235826563197972803556567037646054008
(unsolved after 200 iterations)

Calculation Overload?

It would take a human over an hour to perform the above calculations at a rate of three single digit additions per second, assuming it was done flawlessly.

A program I created (running on my 200 MHz AMD-K6 desktop PC) can perform the above calculations in 0.0006 seconds (0.6 thousandths of a second). It was programming in assembly language - hand coded and optimized in pure computer language - this is almost as fast as it gets without upgrading hardware (I have untested theories on how to improve the algorithm, such as look up tables, multiple digits stored per byte, etc. There's always a way to make it go faster . . .).

This program has continued the above sequence to 32,000,000 iterations. This resulted in a number over 13,000,000 digits long, which has yet to produce a palindrome. It took just over 283 days of calculations on a 266 MHz and a 400 MHz desktop machine, running at separate times. The nature of the algorithm eliminates the option of using multiple machines to improve its calculation speed as each iteration depends on the full answer of the last iteration. (To perform the calculations on multiple machines, you would need a very reliable high speed connection with minimal latency between the machines, and a completely new algorithm that allows partial processing of the huge number. I will not get into it here, but I have some very good ideas regarding this. If anyone is thinking about moving the 196 Palindrome Quest to a parallel network, please take a look at the Processing Across a Network thread, on the 196 Discussion Board. You can see my thoughts there. [Note: This message board is offline. If anyone is willing to host this message board, so it can continue to exist, please contact me. Internet Archive of Processing Across a Network thread.]) To give an indication of how many calculations this actually is, it would take a human being, at the rate of 3 additions per second, 3,300,000 years to accomplish this working 24 hours a day, 7 days a week, without any breaks, and assuming the calculations are done flawlessly.

The next goal was 40,000,000 iterations, which would have resulted in a 16,000,000 digit number taking a little over a year. But, I stopped the quest at this time, since I no longer had access to a computer to continue the calculations. I have passed on my work to Wade VanLandingham. On his web site, 196 and Other Lychrel Numbers, he has continued the quest to over 300,000,000 digits (almost 725,000,000 iterations).

Please take a look at my World's Record page for more information on my involvement in this quest, as well as other palindrome records.

Back to World Records page

 
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Jason Allen Doucette / Xona Games