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 Jason Doucette / Xona Games
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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 + 83596933976833970539 step 43: 177104867844767940077 + 770049767448768401771 step 44: 947154635293536341848 + 848143635392536451749 step 45: 1795298270686072793597 + 7953972706860728925971 step 46: 9749270977546801719568 + 8659171086457790729479 step 47: 18408442064004592449047 + 74094429540046024480481 step 48: 92502871604050616929528 + 82592961605040617820529 step 49: 175095833209091234750057 + 750057432190902338590571 step 50: 925153265399993573340628 + 826043375399993562351529 step 51: 1751196640799987135692157 + 7512965317899970466911571 step 52: 9264161958699957602603728 + 8273062067599968591614629 step 53: 17537224026299926194218357 + 75381249162999262042273571 step 54: 92918473189299188236491928 + 82919463288199298137481929 step 55: 175837936477498486373973857 + 758379373684894774639738571 step 56: 934217310162393261013712428 + 824217310162393261013712439 step 57: 1758434620324786522027424867 + 7684247202256874230264348571 step 58: 9442681822581660752291773438 + 8343771922570661852281862449 step 59: 17786453745152322604573635887 + 78853637540622325154735468771 step 60: 96640091285774647759309104658 + 85640190395774647758219004669 step 61: 182280281681549295517528109327 + 723901825715592945186182082281 step 62: 906182107397142240703710191608 + 806191017307042241793701281609 step 63: 1712373124704184482497411473217 + 7123741147942844814074213732171 step 64: 8836114272647029296571625205388 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step 115: 12080152368339288976183453003554806798828348532619811 + 11891623584382889760845530035438167988293386325108021 step 116: 23971775952722178737028983038992974787121734857727832 + 23872775843712178747929983038982073787122725957717932 step 117: 47844551796434357484958966077975048574244460815445764 + 46754451806444247584057977066985948475343469715544874 step 118: 94599003602878605069016943144960997049587930530990638 + 83609903503978594079906944134961096050687820630099549 step 119: 178208907106857199148923887279922093100275751161090187 + 781090161157572001390229972788329841991758601709802871 step 120: 959299068264429200539153860068251935092034352870893058 + 850398078253430290539152860068351935002924462860992959 step 121: 1809697146517859491078306720136603870094958815731886017 + 7106881375188594900783066310276038701949587156417969081 step 122: 8916578521706454391861373030412642572044545972149855098 + 8905589412795454402752462140303731681934546071258756198 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step 181: 85000782535615151001605670346943251762488426726233875296660620004162653517810047 + 74001871535626140002606669257833262762488426715234964307650610015151653528700058 step 182: 159002654071241291004212339604776514524976853441468839604311230019314307046510105 + 501015640703413910032113406938864144358679425415677406933212400192142170456200951 step 183: 660018294774655201036325746543640658883656278857146246537523630211456477502711056 + 650117205774654112036325735642641758872656388856046345647523630102556477492810066 step 184: 1310135500549309313072651482186282417756312667713192592185047260314012954995521122 + 2211255994592104130627405812952913177662136577142826812841562703139039450055310131 step 185: 3521391495141413443700057295139195595418449244856019405026609963453052405050831253 + 3521380505042503543699066205049106584429448145955919315927500073443141415941931253 step 186: 7042772000183916987399123500188302179847897390811938720954110036896193820992762506 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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.

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 Jason Allen Doucette / Xona Games™