Jason Doucette - World Records - 196 Palindrome Quest, Most Delayed Palindromic Number
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td.smallarial { font-family: Arial, Verdana; font-size: xx-small; }Jason Doucette / Xona.com™location: Arcadia, Yarmouth, Nova Scotia, Canadacontact:[Enable JavaScript for email address]or other forms of contactsocial networks:facebookhi5myspaceforum: Xona ForumsHOME |RÉSUMÉ |PROJECTS/GAMES |GFX |A.I. |TRANSCRIPTS |WORLD RECORDS |WALLPAPERS |CONTACTPROGRAMMING WINDOWS 5TH EDITION ERRATA |DOMAIN HACKS SUGGEST |MATTHEW DOUCETTE |XONA.COM™  Back to Main Resume Page _uacct = "UA-767921-3";urchinTracker();Record AttemptTime CommencedPersonal Achievement1. 196 Palindrome QuestAugust 4, 1999Former WORLD RECORD achieved September 6, 1999.Current WORLD RECORD achieved by Wade VanLandingham.2. Most Delayed PalindromeAugust 13, 1999Former WORLD RECORD achieved September 3, 1999.Current WORLD RECORD achieved November 30, 2005.What is a Palindrome?A palindrome is something that reads the same forward as it does backward.It originated in the early 17th century from the Greek wordpalindromos (palíndromos), literally meaning "running back again."Numbers:Words:Phrases:DNA Sequence:"52125", "4334", "8", and "1758571"."Radar", "I", "Eve", "Deed", and the world's longest in the English language: "Redivider"."Madam, I'm Adam", and the timeless classic: "A man, a plan, a canal... Panama".A segment of DNA in which the nucleotide sequence of one strand mirrors that of the complementary strand.(The world's longest palindromic sentence,based on the timeless classic, "A man, a plan, a canal... Panama",has been created by Peter Norvig.For more word, sentence, and phrase palindromes, please visitJim Kalb's Palindrome Connection.)196PALINDROMEQUEST Numeric PalindromesIn the April 1984 Scientific American "Computer Recreations" column,an article appeared about mathematical patterns(F. Gruenberger, Computer Recreations,"How to Handle Numbers with Thousands of Digits, and Why One Might Want To.",Scientific American, 250 [No. 4, April, 1984], 19-26.).Here's the algorithm:Pick a number.Reverse its digits and add this value to the original number.If this is not a palindrome, go back to step 2 and repeat.Do all numbers eventually become palindromes by this process?It was suggested that this is the case.ExamplesMost numbers become palindromes fairly quickly, in only a couple of steps:1313 + 31 = 446464 + 46 = 110110 + 011 = 1218787 + 78 = 165165 + 561 = 726726 + 627 = 13531353 + 3531 = 4884In fact, about 80% of all numbers under 10,000 solve in 4 or less steps.About 90% solve in 7 steps or less.A rare case, number 89, takes 24 iterations to becomea palindrome. It takes the most steps of any number under 10,000 that has beenresolved into a palindrome:8989 + 98 = 187187 + 781 = 968968 + 869 = 18371837 + 7381 = 92189218 + 8129 = 1734717347 + 74371 = 9171891718 + 81719 = 173437173437 + 734371 = 907808907808 + 808709 = 17165171716517 + 7156171 = 88726888872688 + 8862788 = 1773547617735476 + 67453771 = 8518924785189247 + 74298158 = 159487405159487405 + 504784951 = 664272356664272356 + 653272466 = 13175448221317544822 + 2284457131 = 36020019533602001953 + 3591002063 = 71930040167193004016 + 6104003917 = 1329700793313297007933 + 33970079231 = 4726708716447267087164 + 46178076274 = 9344516343893445163438 + 83436154439 = 176881317877176881317877 + 778713188671 = 955594506548955594506548 + 845605495559 = 18012000021071801200002107 + 7012000021081 = 8813200023188The 196 Palindrome Quest (a.k.a. 196 Algorithm, 196 Problem)Does every number eventually become a palindrome?Nobody knows for sure, since it has never been proven.There are some numbers that do not appear to ever form a palindrome.The first one is 196.Such numbers are calledLychrels.The search to resolve this number has been referred to as the 196 Algorithm or the 196 Problem,but normally called the 196 Palindrome Quest.There are a few references to the 196 Palindrome Quest before the 1984 issue of Scientific American.Two references were found from a Math Central page:Heiko Harborth, On Palindromes, Mathematics Magazine, (1973) 96-99C.W. Trigg, Palindromes in addition, Mathematics Magazine, 40 (1967) 26-28.The 1973 paper states that Harborth said that Trigg checked all integers less than 10,000 in 1967 and found that 249seemed to never form a palindrome. 196 would be the first of those 249 numbers.John Walker's Three Years Of Computing pagestates that 196 had originally been iterated over 10,000 times without yielding a palindrome.Later, this number had been carried through 50,000 reversals and additions byPaul C. Leylandyielding a number of more than 26,000 digits without producing a palindrome.Paul explains his program in an email toWade VanLandinghamon August 19, 2002:"The work you refer to was done about 20 years ago on a 4 MHz Z80-based machine running CP/M.The core reverse&add and the palindromicity detector were written in assembler and the I/O etc was written in Algol-60.The machine only had 32K of memory (actually rather a lot for those days) and I ran the program until it ran out of memory --- which explains the limit chosen for the number of iterations."- Paul C. LeylandAgain, P. Andertoncontinued the process up to 70,928 digits (170,000+ iterations) without encountering a palindrome.(This Google Groups postingclaims a 70,000 digit result was accomplished in 1987,so perhaps this was P. Anderton's effort.)The First Iterations of 196Here is 196 iterated 200 times for those of you who are curious.Three Years of Computing (196 taken to 1,000,000 digits)On August 12, 1987,John Walkercommenced a program he created on a Sun 3/260 workstation(using a Motorola 68020 microprocessor)to take the search even further.Five minutes before midnight, on May 24, 1990, almost three years later,his program ended after 2,415,836 iterations,yielding a number 1,000,000 digits long - with no palindrome in sight.The next day, John Walker made this 1,000,000 digit number and his program availableto people on a page on the Internet,Three Years Of Computing,for any who would like to continue his quest without redoing his three years of work.About Two Months of Computing (from 1,000,000 to 2,000,000 digits)With a interest in the palindrome quest since the 1984 issue of Scientific American,Tim Irvin found himself with access to a supercomputer,a Concurrent Computer Corporation Maxion model 9502,in 1995.Searching the Internet for work already completed on the palindrome quest,he found John Walker's web page.He 'borrowed' his program,and with the help of fellow programmer Larry Simkins,they put it to work again to continue the quest from the 1,000,000digit number.On August 22, 1995, after about two months of calculations,the program stopped after having calculated 196 to 2,000,000 digits.With a difference of 8 years in computer generations,Tim's supercomputer achieved three times the amount of calculations(as required for John Walker's three years of work)in only two months.This number is available off of Tim's web page,About Two Months of Computing,for any person wishing to continue the quest themselves.Continuing the Quest (to 13,000,000 digits and beyond)I have never even seen a supercomputer, so why am I continuing the quest?Because I created an assembly language program that can perform the necessarycomputations on an average desktop computer faster than the unoptimized programspreviously used on supercomputers can.(Please note that the original programscreated to compute the 196 Palindrome Quest were made to be memory efficient,at the sacrifice of speed. Despite these concerns,I found that these programs missed certain optimizations that could have savedthe same amount of memory with no speed sacrifices, as well as certainoptimizations that had no bearing on memory use whatsoever).My program started calculating the 196 Palindrome Quest on Monday, August 9, 1999 ona Pentium II 266 MHz PC.The program reached the 1,000,000 digit mark in 1 day and 18 hoursI am going to retest this time at some point,since my code was bugged, causing it write a total of 54 megabytes of junkto the screen and a data file, which slowed the calculations.The program reached the 2,000,000 digit mark in an additional 5 days and 10 hours.Both these numbers check out with John Walker's and Tim Irvin's work.I am 100% confident that my program is bug-free, I am going tocontinue the work for as long as I have access to idle time of a computer.The program reached the 3,000,000 digit mark in an additional 8 days and 7 hours.The program reached the 4,000,000 digit markin an additional 8 days and 14 hours.This was achieved on Monday, September 6, 1999,breaking the previous record of 3.9 million digits(according to the rec.puzzles archive),4 days quicker than expected, thanks to the help of a Celeron 400 MHzmachine, on which the program ran for some of its processing.The program reached the 5,000,000 digit mark in an additional 13 days and 8 hours.The program is now running solely on the Celeron 400 MHz processor.The program reached the 6,000,000 digit mark in an additional 14 days and 4 hours.The program reached the 7,000,000 digit mark in an additional 26 days and 12 hours.The program reached the 8,000,000 digit mark in an additional 33 days and 23 hours.The program reached the 9,000,000 digit mark in an additional 59 days and 12 hours.(The increased length of time for this million digits is probably caused by the fact that theprogram was stopped and continued very many times during this period.)The program reached the 10,000,000 digit mark in an additional 34 days and 1 hour.The program reached the 11,000,000 digit mark in an additional 26 days and 3 hours.The program reached the 12,000,000 digit mark in an additional 28 days and 5 hours.The program reached the 13,000,000 digit mark in an additional 28 days and 23 hours.(The program had a little help of a Pentium III 450 MHz machine for this million digits,although the program can only be run on one machine at any given time.)My 13,000,000 digit mark is published in the November / December 2001 Issue of Yes Mag: Canada's Science Magazine for Kids: Click to enlarge [75 dpi resolution image = 154 Kb]Click to enlarge [150 dpi resolution image = 452 Kb]The program was averaging over 12,000,000 single digit additions per secondwith the Celeron 400 MHz machine(i.e. it can add two 12,000,000 digit numbers in one second).That's over 1 trillion (1,000,000,000,000) single digit additions per day.Other Work - Ian J. PeterAfter my correspondence with Ian J. Peter,who is currently calculating all integers up to 2,000,000,000 out to 200,000 digits each,he started to improve his palindrome program. His hand optimized assembly program,running on an AMD Athlon 500MHz system runs from 4 to 8 times as fast asmy program on an Intel Celeron 400MHz.Why such a huge speed increase?Several factors. His program has been largely optimized algorithmically,as well as in hand optimization at the assembly level -but I am not goingto give out any secrets here; it is not for me to release, so please ask him instead.But I can say a few technical notes:My program uses flat-mode (unreal-mode) 32-bit addressing in 386-code,using Turbo Pascal 7.0, due to my lack of a compiler that allows trueprotected mode memory addressing.This is amusing, given that the Intel 80286 (i.e. 286) processorintroduced this feature (My old Tandy 1000 SX had one of those).Ian is using protected-mode 32-bit addressing,running under Linux, and is taking advantage of thenew instruction set on the newer machines.As far as hardware is concerned, this is the firsttime AMD has impressed me. I have an AMD K6-200MHz,which I am not impressed with.My 3Dfx VooDoo2 card sits and waits for the processorto hand it information.But, the Athlon is another story. It has 256 KB L1 cache (more L1 cache thanCelerons have in L2 cache), which is 4 times the amount of regular Pentium III systems.It has 512 KB L2 cache, butcan have as much as 8,192 KB.This does not mean you should go out and buy one.You should buy the processor that will benefit your type of work most.The nature of our palindrome programs - given thatthe code is very small, and that memory accessis always in sequential order - benefit from the Athlon architecture even more so thannormal applications or games.His program reached 1,000,000 digits in just over 5 hours,and reached 10,000,000 digits in just over 30 days, running on hisdedicated 500 MHz Athlon.Other Work - István BozsikIstván Bozsikfound the same previous work from other people, mentioned above,that I did. It sparked his interest, and he quickly found that he couldalso make a program to run the 196 Palindrome Quest much faster than they did.He wrote his program on May 7, 2000.On hisweb pagehe shows that he took the quest to 6 million digits.Unfortunately, I had already passed 6 million digits with my own work.István had not been able to locate my web page via search enginesuntil after his achievement.His page has lots of links to other web pages regardingpalindromic numbers.Work Passed On - Wade VanLandingham(to 300,000,000 digits and beyond)I no longer have the processing time to continue the quest.I have given my 13.5 million digit record toWade VanLandingham.He has taken this number, and continued it toover 300 million digits (almost 725 million iterations).He has compiled a lot of information from all kinds of sourcesregarding the 196 Palindrome Quest on his web page,196 and Other Lychrel Numbers.It is worth a visit.196 Discussion - Message Board[Note: This message board is offline.If anyone is willing to host this message board, so it can continue to exist,please contact me.]Felipe Barone has created amessage board for discussionthe 196 Palindrome Quest.You may like to take a look around.I have some interesting thoughts regarding computation of the 196 Palindrome Queston a network in the following thread:Processing Across a Network.[Internet Archive ofProcessing Across a Network thread.]FRIDAY, JANUARY 27, 2006 UPDATE:It has been done!Pierre-André Laurent has created a networked 196 Palindrome Quest application:196 and Other Lychrel Numbers - Distributed Software ComparisonsMOSTDELAYEDPALINDROMICNUMBER The Previous RecordWith the exception of my work, the only information on theMost Delayed Palindromic Numberwas found at Ian J. Peter'swebsite:Search for the Biggest Numeric Palindrome.Before his work, the most delayed palindromic number that was known was10,911.It takes55 iterationsto become a palindrome that is 28 digits long.After his extensive searching on all numbers from 1 to 9,999,999, he has foundthe following results:Previous Records - found by Ian J. PeterNumberIterationsResulting Palindrome147,996150,2961,000,6891,005,7441,017,5017,008,8999,008,2995864787980829688344533248416747614842335443886820495694655501210555649659402867965898843249669456465496694234889856977965898843249669456465496694234889856971467444396014326533335623410693444764168586378655656964999946965655687368586555458774083726674580862268085476627380477854555The ProcedureIan Peter tested all of his numbers to 200,000 digits (~500,000 iterations).He proved that there is no number from 1 to 9,999,999 thatforms a palindrome in over 96 iterations, but in under ~500,000 iterations.Using this information, I set my program (the same program currently tryingto break the 196 Palindrome Quest record) to retrieve the number of iterationsrequired to form a palindrome for all numbers from 0 to 9,999,999. This wasdone by setting a limit of 96 iterations. If the number did not resolve into apalindrome after this many iterations, it was marked as 'infinite'. Becauseof Ian Peter's work, I know that I would have to take these numbers toat least 500,000 iterations before resolving them. Knowing that the likelihoodof this happening is very slim, I decided not to take them any further.After analyzing the information, I determined that the chances of thenext most delayed palindromic number being over 255 iterations, withoutfirst finding one that resolves in fewer iterations, was extremely unlikely.As a result of this, I determined the best way to break the record of theMost Delayed Palindromic Number was to continue from 9,999,999 on,looking at a maximum of 255 iterations. By limiting the number of iterationsto such a small number, an unbelievable amount of time is saved(If I wished to double the iteration limit to 510,the process instantly becomes four times slower).Using this method, within the first two days of running the program,my program had solved two new records:100,239,862, which solves in 97 iterationsand140,669,390, which solves in 98 iterations.However, I did not realize until after speaking withIan J. Peterthat his program had already solved these records, and he simply had not updated his web pageto show the results.Here are Ian J. Peter's latest results:Previous Records - found by Ian J. PeterNumberIterationsResulting Palindrome100,239,862140,669,3901,090,001,9211,009,049,4071,050,027,9481,304,199,6931,005,499,52697989910110410510913454289533677631256753655556357652136776335982454311345428953367763125675365555635765213677633598245431663454444878830167588644688576103887844445436615434342665875551147797222797741155578566243434515831124885795990016569666669656100995975884211385583112488579599001656966666965610099597588421138566330069478378985774345546664554347758987387496003366Thoughts on Improving the SearchThe current procedure is not thefastest way to find the most delayed palindromic number.Most numbers being checked are 'consequences' of numbers already checked.For instance, a few consequences of140,669,390 are:150,669,380,160,669,370,170,669,360,180,669,350,190,669,340.These were found by simply changing the secondand the second-to-last digits.Note the pattern - each of these numbers, after their first iterationwill all become the same number(the two digits being modified sum to 13 each time),and thus will eventually allyield the same palindrome. It does not make sense to recalculate thesesame iterations over and over again.Even more consequences can be found by changing the third and the third-to-last digits, and so on.These are all known as 'first order consequences' since they producethe same number after just one iteration.Other consequences existthat produce the same number after two iterations(these are called 'second order consequences' since they producethe same number after two iterations),and after three iterations('third order consequences'), and so on.Although, these are a little harder to find. If you'll lookat Ian Peter's work in the table above, you will notice some of his resultsform the same palindrome as others. They are consequencesof each other.By messing with the digits, as we did above,we can find 1 * 6 * 4 * 4 = 96 numbers, including the original,that all have the same properties as 140,669,390.Almost every number, not just ones that become palindromes,have lots of first order consequences of itself- and in most cases, more than just 96.However, until I write a program to fix these problems,the program will continue to run on its machine, for aslong as I have access to it. It is currently calculating100,000,000 numbers per day - so, hopefully we have seen justthe beginning.Improving the SearchFRIDAY, AUGUST 20, 1999 UPDATE:BAD NEWS: I have lost access to the computer that was dedicated to thesecalculations.GOOD NEWS: After a little bit of thinking, I have 'discovered' an algorithm inwhich will minimize repetitive calculations that are caused by first order consequencesof the same number. This is an exponential improvement on my previous algorithm.Allowing my computer 2 weeks of work with myold algorithm, at its current rate, would'solve', to 255 iterations, all numbers from 0 to 1,360,000,000.In others words, all 9-digitnumbers and more. Theoretically, mynew algorithm, given the same numberof calculations (2 weeks of computations on the same machine),could 'solve', to 255 iterations,all numbers from 0 to 99,999,999,999,999.In others words, all 14-digit numbers. (My previous algorithm would take 2,800 years to do this)I will probably have toincrease my iteration limit as the numbers grow large,thus taking more time than just 2 weeks,but the above is just to give an indicationof the improvement of the algorithm itself.Enough talking - ina few days I am going to implement the algorithm, and hopefullyit lives up to what I expect. I will run it on my home computer, which should'see' about 50% of the processing that it was gettingon the dedicated computer I used to have.Implementing the SearchFRIDAY, SEPTEMBER 3, 1999 UPDATE:I have reprogrammed my new exponential algorithm to find the smallest numberthat solves into a palindrome for each distinct iteration count. For example,it will find that 10,905,963 solves in 71 iterations,regardless that there is a smaller number thatsolves in more iterations(9,008,299 in 96 iterations).10,905,963 is reported because it is the smallest number that resolves in 71 iterations,out of all numbers that resolve in 71 iterations.Thus the program not only reports numbers thatbreak the record for more iterations, but also reports the smallest number that solves in x iterations,for all x.The program adjusts the iteration limit so that it is three times that of thecurrent record. This was programmed after analyzing the data alreadyaccumulated, and it was determined that the chances of finding a numberthat solves in more than three times the number of iterations of the current record,without first finding one that solves in less than three times the numberof iterations of the current record, is extremely slim.I have started the new program, calculating all most delayed palindromic numbersfrom scratch on Friday, September 3, 1999 on my AMD-K6 200 MHz computer.FRIDAY, SEPTEMBER 3, 1999 UPDATE:Within an hour and a half(1 hour, 29 minutes, 45 seconds, to be exact)of started the new program from scratch,it had solved a new world record:The 11 digit number10,287,799,930resolves into a 65 digit palindrome in 134 iterations.Amazingly, only 3 minutes later, a new world record was found:The 11 digit number10,700,572,940resolves into a 66 digit palindrome in 149 iterations.FRIDAY, AUGUST 4, 2000 UPDATE:More statistics of the program follows:Jason's New Exponential AlgorithmPerformance (AMD-K6 200 MHz)It solved all 6 digit numbers and below in 1 second.It solved all 7 digit numbers in 28 seconds.It solved all 8 digit numbers in 1 minute 38 seconds.It solved all 9 digit numbers in 21 minutes 51 seconds.It solved all 10 digit numbers in 59 minutes 16 seconds.It solved all 11 digit numbers in 20 hours, 29 minutes.It solved all 12 digit numbers in 1 day, 20 hours.It solved all 13 digit numbers in 31 days, 9 hours.It solved all 14 digit numbers in 63 days, 9 hours.It solved 8.34% of all 15 digit numbers.FRIDAY, AUGUST 4, 2000 UPDATE:During this, it solved the following world records,each beating the last record my program set:New World Record: The 13 digit number 1,600,402,195,090 resolves after 188 iterations.New World Record: The 15 digit number 107,000,020,928,910 resolves after 192 iterations.New World Record: The 15 digit number 100,120,849,299,260 resolves after 201 iterations.At this point, the program stopped due to lack of a computer to run the program.The ResultsMONDAY, JUNE 17, 2002 UPDATE:I have not been able to run my program since Friday, August 4, 2000.At the time, I only had 8.34% of all the 15 digit numbers solved.I could not continue because the program was written in DOS underFlat Mode, which refuses to run under Windows 98 or greater.These operating systems no longer allow the computer to be (easily) rebootedinto a fully DOS compatible mode, which is not a Windows Shell.On Sunday, May 19, 2002, I finally reprogrammed my algorithm in Windowsto allow the search to continue. It is too bad that I have waited this long,because I would be far into the 17 digit numbers right now.After some initial beta-testing, I started the new programagain from scratch on Thursday, March 30, 2002. It has sincereached, and surpassed the numbers checked of my old DOS program.Note that it is only running part time on my AMD 1700+ XP processor.The records, including the World Record for Most Delayed Palindromic Number(highlighted in red),are listed in the below table. Click on any of the numbers to pop-up a calculationpage that proves its result.WEDNESDAY, APRIL 16, 2003 UPDATE:I am now into the 17 digit numbers. This is going to be an interesting set ofnumbers to look at, since after the result of the 16 digit numbers, there is onlyone iteration count below thecurrent (as of April 16, 2003)world record of 201 iterations(the 15 digit number 100,120,849,299,260)that has yet to solve out: 160.Any new information to be reported by the program will either be the 160 iterationresult, or a new world record.We can guess at what the 160 iteration result would be, by iterating thesmallest number to resolve in 161 iterations (the 16 digit number 6,000,000,039,361,479)a single time.Thus we can tell that the 17 digit number 15,741,639,339,361,485will solve out in 160 iterations.Its lowest first order consequence('consequences' are explained above on this page)is 14,200,033,399,975,995.Therefore we will expect to see a result for 160 iterations which is, at largest, this 17 digit number.TUESDAY, APRIL 29, 2003 UPDATE:I have incorporatedBenjamin Despres'reverse-and-add code into my algorithm for findingMost Delayed Palindromic Numbers, as his code is approximately three timesfaster than my reversal-addition code.The core exponential algorithm that determines which numbers to testremains unchanged. I have started the quest from scratch, again,to ensure that his code is operating perfectly with my code.I have already extensively tested his code with extreme casesto ensure its accuracy. The biggest problem that I had porting his codeinto my program is that his code was not designed to be re-run multiple times.He programmed the initialization portion of it for a one-shot deal.Initialization code is normally left unoptimized as it is onlyrun once, and therefore does not matter for performance. It onlyaffects the start up time of a program, and for a process that takesseveral months (his code was computing the 196 Palindrome Quest),it does not matter if the initialization code takes 1/1,000th of a secondor 1/10,000th of a second. I re-codedthe initialization section to re-initialize only the parts of memorythat has to be for optimal performance.WEDNESDAY, JUNE 2, 2003 UPDATE:My old code had solved 0.297% of all 17 digit numbers.My new program(with Benjamin Despres' reversal-addition code)finally surpassed this today.All results match prior results.A few hours later, my program also finally found a new world record since theold recordwas found solving the 15 digit numbers.My old program was within hours of solving this new record,and was delayed for almost two monthsdue to the restarting of the program withBenjamin Despres'reversal-addition code.WEDNESDAY, JUNE 2, 2003 UPDATE:Today, at 2:47 PM, my program(with Benjamin Despres' reversal-addition code)solved a new world record:The 17 digit number10,078,083,499,399,210resolves into a 112 digit palindrome in 233 iterations.This new world record beats the old world recordthat has stood for about 3 years(set by my program sometime between November 1999 to August 2000;the exact date was never recorded)which was the 15 digit number100,120,849,299,260which resolves into a 92 digit palindrome in 201 iterations.Also, later that day, at 7:28 PM, a record for 160 iterations was found:10,019,017,999,499,510.It is lower than the number that we expected could be the potential record(14,200,033,399,975,995),and it is on a different thread (series of numbers that are formed by the reversal-addition iteration),as the resultant palindrome is different from that ofthe record for 161 iterations.(Please note that on January 28, 2004, an even smaller number for 160 iterations was found,on yet another thread:10,000,000,730,931,027.)THURSDAY, JULY 10, 2003 UPDATE:Today, at 12:49 PM, my program(with Benjamin Despres' reversal-addition code)solved a new world record:The 17 digit number10,442,000,392,399,960resolves into a 111 digit palindrome in 236 iterations.WEDNESDAY, DECEMBER 15, 2004 UPDATE:At 10:44 AM, after exactly 500 days of processing from scratch, my program(with Benjamin Despres' reversal-addition code)finally resolved all 17 digit numbers.The majority of that time (444 days of those 500) was required for just the processing of the 17 digit numbers.Due to the optimizations in my algorithm, it does not iteratively check each number.My algorithm determines which numbers can be eliminated from the search and still maintain 100% accurate results.As a result, my program actually only checked 186,819,193,449 total numbers,instead of 99,999,999,999,999,999 numbers,to compute the results of all 17-digit numbers and smaller,which is a significant optimization.It would have taken over 700,000 years, on the same computer,to compute all these numbers without this optimization.MONDAY, MARCH 21, 2005 UPDATE:Vaughn Suite has corresponded with me regarding my quest over the past few weeks.He suggested performing a statistical analysis of the data, to determine the likelihoodof missing a record at any given iteration limit.Using such analysis, we can set the limit to match our preferred risk.If this shows the current limitation is overkill,then we will speed up the program by reducing the limit.Originally, I took a naive approach and simply set the limit to be 3 times that of the last record found.This was done only because, at a quick glance, it appeared to suffice.It appeared quite a bit less safe than Ian Peter's limitation of 200,000 digits(about 500,000 iterations, which varies slightly depending on the number checked).I should again note thatIan Peter's workhas exhaustively shown that numbers that do not resolve quicklyseemingly never resolve.This has been crucial in allowing us to use the below statistical analysis on my data.Vaughn Suite's analysis started with the most delayed palindromic number for each digit length, as follows(updated Sunday, September 25, 2005 upon completion of the 18-digit set):Largest Delay per Digit SetDigit SetLargest Delay1-digit numbers2-digit numbers3-digit numbers4-digit numbers5-digit numbers6-digit numbers7-digit numbers8-digit numbers9-digit numbers10-digit numbers11-digit numbers12-digit numbers13-digit numbers14-digit numbers15-digit numbers16-digit numbers17-digit numbers18-digit numbers22423215564969698109149149188186201197236232You will note that this data is not immediately known from therecords I show on this page.My Most Delayed Palindromic Number queststores only the smallest number that resolves at each iteration depth.Therefore if a record is set in a smaller digit set,any other numbers that resolve in the same amount of iterations fromlarger digit sets will not be recorded, since they are larger numbers.These numbers are not recorded even if they are themost delayed palindromic number of that digit set, simply becausethis was not the purpose of the program.For example, in the 7 digit set, the most delayed palindromic number is 9,008,299 which resolves in 96 iterations.There are two (consequences not counted)8 digit numbersthat solve in 96 iterations:15,002,893and15,059,593,but they are not recorded.The most delayed palindromic number recorded for the 8 digit set is10,309,988 which solves in only 95 iterations.The two that resolve in 96 iterations are not smaller than the7 digit number,9,008,299,that resolves in 96 iterations, and my program saves only the smallest one.Vaughn Suite verified these maximum iteration depths for each digit sethimself for all digit sets up to 13 digits,and got the maximum depths for the 15 digit and 17 digit sets from this page,and the for the 14 digit and 16 digit setsfrom my notes onWade VanLandingham's site(under Other People's Notes).He noted that the values appeared to follow a linear trend,estimated the maximum iteration depth for the 18 digit setwould likely be 250 and suggested testing to 350 iterations,instead of multiplying the last previous record by three.After some discussion, we performed linear regression analysis,which yielded the following equation:Expected Maximum Iteration Depth = 14.416667 * Digit Length - 18.338235The standard deviation was found to be 11.245233.We concluded that we could use normal distribution to describe the data.There is a 98.83% correlation between these iteration limits and the digit length of the set they represent.(If you are a statistician, and believe this is incorrect, pleasecontact me.)Armed with this information,we did further spreadsheet analysis to determine the probabilityof a number being missed for the 18 digit set,giving any iteration limit.We could also input the probability (risk) level we were prepared to accept,and the sheet would inform us of the iteration level required for such a risk.In retrospect, I was doing far more calculations than I needed to.Using a limitation of 708 iterations for the 18 digit set representsa missed record would have to be 41.51 standard deviations away from the mean.This is an astronomically small chance. I cannot even calculate the chanceswithin Excel due to the precision limitations of the program.(If someone hasMaple,Mathematicaor anotherComputer Algebra System,pleasecontact me,and perhaps we can compute the actual amount.)To give you an idea just how crazy this is, 68% of all numbers fall within 1 SD (standard deviation) of the mean,95% fall within 2 SD, 99.7% fall within 3 SD, 99.993% fall within 4 SD,99.99994% fall within 5 SD, 99.9999998% fall within 6 SD, 99.9999999997% fall within 7 SD, etc.In another perspective, if we wish to have a 1 in 1,000 chance of missing a record, we'd set the iterationlimit 3.09 SD away from the mean. To have a 1 in 1,000,000 (one million) chance of failure,we'd set the iteration limit 4.75 SD away from the mean.For 1 in 1,000,000,000 (one billion), set it 6.00 SD away.For 1 in 1,000,000,000,000 (one thousand billion), set it 7.04 away.For 1 in 1,000,000,000,000,000 (one million billion), set it 7.94 away.Imagine what the chances of failure would be going all the way to 41.51!Thus, the quest can now be sped up by using this information.After much discussion, I personally determinedthat a 1 in 10,000,000 chance of missing a record was reasonable.This is based on my personal conjecture that when a new world record is found,there are numerous other records found for iterations just below the world record.In other words, a world record has yet been found in which there exists no otherrecords for iteration depths almost as deep as the record.Therefore, in the unlikely (1 in 10,000,000 chance) eventthat we will miss a record, we will likely know that we missed it,due the likelihood of numerous other numbers resolving in iterationdepths very close to the limit.In this case, we can increase the iteration depth, and re-test the entire data set.Yes, it would be depressing to have to redo this work,but I believe in the 1 in 10,000,000 chance that this happens,it is worth the speed up in the program.For a 1 in 10,000,000 chance of missing a record, our formulas determined an iterationdepth of 299.63 is required, therefore I set the depth to 300 for the 18 digit set.The actual chance of missing a record for a depth of 300 iterations is 1 in 11,921,892,for those who are curious.I would like to extend a great thanks to Vaughn Suite for his help.Using this information, the Most Delayed Palindromic Number questhas been sped up by over 5.5 times.SUNDAY, SEPTEMBER 25, 2005 UPDATE:On Sunday, September 25, 2005, at 3:16 am,my program(with Benjamin Despres' reversal-addition code)completed the 18-digit number set.The most delayed palindromic 18 digit number solves in 232 iterations,as you can see by the above table labelled 'Largest Delay per Digit Set'.Using this information to update the statistical analysis,we arrive at the following new formula:Expected Maximum Iteration Depth = 14.255934 * Digit Length - 17.320261The standard deviation was found to be 11.087996.There is a 98.96% correlation between these iteration limits and the digit length of the set they represent.Using this information, for a 1 in 10,000,000 chance of missing a record,the iteration limit should be set to 311.20. I have rounded this value up to 315for the 19-digit set. A limit of 315 iterations actually represents a1 in 66,981,399 chance of missing a record.Certain numbers do not resolve into a palindrome within the limit tested,and we believe such numbers will never solve, no matter how many iterations they are taken to.Such numbers are called Lychrel numbers.The percentage of Lychrels that occur increases with each new digit length tested:Percentage of LychrelsDigit Length% Not Resolved1-digit numbers2-digit numbers3-digit numbers4-digit numbers5-digit numbers6-digit numbers7-digit numbers8-digit numbers9-digit numbers10-digit numbers11-digit numbers12-digit numbers13-digit numbers14-digit numbers15-digit numbers16-digit numbers17-digit numbers18-digit numbers0.00%0.00%1.67%3.51%7.25%14.45%22.17%31.30%40.42%49.61%57.82%65.44%71.64%77.17%81.41%85.22%88.03%90.55% (Please note the above data is obtained from the sub-set of numbers my program tests.My program does not test every number, due to the optimizations explained earlier.So, while this graph is not 100% accurate, it is certainly very close. Most likely, itis accurate to well within the precision shown.)CURRENT WORLD RECORDWEDNESDAY, NOVEMBER 30, 2005 UPDATE:Today, at 5:20 AM, my program(with Benjamin Despres' reversal-addition code)solved a new (current) world record.The 19 digit number1,186,060,307,891,929,990resolves into a 119 digit palindrome in 261 iterations.Proof: View the reversal-addition sequence for 1,186,060,307,891,929,990This new world record beats the old world record set by my program(with Benjamin Despres' reversal-addition code)2 years, 143 days ago,which was the17 digit number10,442,000,392,399,960which resolves into a 111 digit palindrome in 236 iterations.(I wasn't the first to beat my July 10, 2003 record of 236 iterations.A program written by Vaughn Suite found the 19-digit number1,000,000,079,994,144,385,which resolves into a 119 digit palindrome in 259 iterations, on July 26, 2005 at 8:27 AM.)Most Delayed Palindromic Number Records'Zero Iterations' Note:These numbers are calculated according to the number of iterationsof reversal-addition required to reach a palindrome.One iteration, at minimal, is performed.Thus, if the number is already a palindrome, it is not considered to be palindromic after 0 iterations.As a result, we have examples such as1,5 and999.'Solved' Note:In the graph below, 'solved' means that the numbers were iteratedto a calculated maximum iteration needed to show within a reasonable doubtthat the number will never solve.For all 17 digit numbers and smaller,I determined it was reasonable to iteratethe numbers to three times the amount of the current world record.For example, if the record is 200 iterations,my program will test other numbers to 600 iterations.This has been shown to be significant overkillin light of the statistical analysis.For the 18 digit set,Vaughn Suite and I used statistical analysisto determine a reasonable limit of 300 iterations.This limitation has a chance of less than 1 in 12,000,000 that we will miss a record.For the 19 digit set,using the same statistical analysiswe determined a reasonable limit of 315 iterations.This limitation has a chance of less than 1 in 67,000,000 that we will miss a record.It should be noted that Ian Peter's resultshave been essential for the assumptions made in these calculations.His extensive search performed on all 9 digit numbers and below(taking them all to 200,000 digits [slightly less than 500,000 iterations])has shown, beyond any reasonable doubt,that if a number does not resolve into a palindrome quickly,it will never resolve into a palindrome.Although it is believed that these numbers never solve out, like196,it may be impossible to confirm.Until it can be proven that they will never solve,the following results are only conjectures.Solved all 1 digit numbersNumberIterationsDigitsResultant Palindrome151212211Solved all 2 digit numbersNumberIterationsDigitsResultant Palindrome59697989346244451311114884440448813200023188Solved all 3 digit numbersNumberIterationsDigitsResultant Palindrome166188193829167849177999739989869187578101114151617192223566881010101011131345254233332233332885555888855558888368863888836886388893977939852333333258954000459888132000231888813200023188Solved all 4 digit numbersNumberIterationsDigitsResultant Palindrome1,3972,0691,7971,7986,9991,2979121318202188101114138855558852788725883688638889540004598166684884866618813200023188Solved all 5 digit numbersNumberIterationsDigitsResultant Palindrome10,79710,85310,92110,97113,29710,54813,29317,79320,88980,35913,69710,79415,89170,75970,26910,67710,83310,9112526272829303132333738394047525354551616161618171717172222222226282828281676404554046761445559744795554444555974479555448802202552022088893974888888479398178587688867858711785876888678587144035358885353044440353588853530446839849878998789489386683984987899878948938668321236953359632123866832123695335963212386145257565444994445657525414668731596684224866951378664466873159668422486695137866446687315966842248669513786644668731596684224866951378664Solved all 6 digit numbersNumberIterationsDigitsResultant Palindrome700,269106,977108,933600,259131,996600,279141,996600,579147,996178,992190,890600,589150,29634353645465051575859606364222222262628283131313133336832123695335963212386683212369533596321238668321236953359632123861452575654449944456575254114525756544499444565752541466873159668422486695137866446687315966842248669513786648834453324841674761484233544388883445332484167476148423354438888344533248416747614842335443888834453324841674761484233544388682049569465550121055564965940286682049569465550121055564965940286Solved all 7 digit numbersNumberIterationsDigitsResultant Palindrome1,009,2271,007,6191,009,2461,008,6281,007,3771,001,6991,009,1501,058,9211,050,9951,003,5691,036,9741,490,9913,009,1791,008,5951,064,9121,998,9997,008,4291,000,6891,005,7441,007,6017,008,8999,008,29941424344484956616265666768697075777879808296232223232827283233323232353535323839393838486834449727969727944438615565342872278243565514514445443202344544415448852787646664678725884883674699729922992799647638816897794447942497444977986168421656644286682446656124861896633685246796697642586336698168204956946555012105556496594028614758724578598888889587542785741147587245785988888895875427857411475872457859888888958754278574146563056797844547874544879765036564465630567978445478745448797650365644656305679784454787454487976503656415521561387579888897578316512551146744439601432653333562341069344476417965898843249669456465496694234889856977965898843249669456465496694234889856971467444396014326533335623410693444764168586378655656964999946965655687368586555458774083726674580862268085476627380477854555Solved all 8 digit numbersNumberIterationsDigitsResultant Palindrome10,905,96310,069,78510,089,34211,979,99010,029,37210,029,82616,207,99090,000,58910,309,988717273747681839495383232323938384848356954879767784335885334877679784596531552156138757988889757831651255115521561387579888897578316512551155215613875798888975783165125517965898843249669456465496694234889856976858637865565696499994696565568736858668586378655656964999946965655687368586555458774083726674580862268085476627380477854555555458774083726674580862268085476627380477854555Solved all 9 digit numbersNumberIterationsDigitsResultant Palindrome100,389,898100,055,896110,909,992160,009,490800,067,199151,033,997100,093,573103,249,931107,025,910180,005,498100,239,862140,669,39084858687888990919293979845414040464646464648525258422798878766858919924299198586678788972248588682199585544879735653797844558599128688352569827522026889722798862022572896525335256982752202688972279886202257289652535852497685678899643696996963469988765867942585585249768567889964369699696346998876586794258558524976856788996436969969634699887658679425855852497685678899643696996963469988765867942585585249768567889964369699696346998876586794258555545877408372667458086226808547662738047785455513454289533677631256753655556357652136776335982454311345428953367763125675365555635765213677633598245431Solved all 10 digit numbersNumberIterationsDigitsResultant Palindrome1,090,001,9217,007,009,9091,009,049,4079,000,046,8991,050,027,9481,304,199,6935,020,089,9491,005,499,5269910010110310410510810946494949494953536634544448788301675886446885761038878444454366154343426658755511477972227977411555785662434345115434342665875551147797222797741155578566243434515831124885795990016569666669656100995975884211385583112488579599001656966666965610099597588421138558311248857959900165696666696561009959758842113856633006947837898577434554666455434775898738749600336666330069478378985774345546664554347758987387496003366Solved all 11 digit numbersNumberIterationsDigitsResultant Palindrome10,000,505,44810,000,922,34710,000,696,51110,701,592,94310,018,999,58310,000,442,11910,000,761,55410,084,899,97010,006,198,25018,060,009,89011,400,245,99616,002,897,89218,317,699,99037,000,488,99910,050,289,48590,000,626,38910,000,853,64813,003,696,09310,050,859,27110,287,799,93010,000,973,03710,600,713,93310,942,399,91160,000,180,70911,009,599,79616,000,097,39210,031,199,49410,306,095,99110,087,799,57010210610711011111211311411511611711811912212313013113213313413513613714414514614714814949505653565656565656565656585865656565657171716666666666665831124885795990016569666669656100995975884211385157399297003411133024269779624203311143007929937511332389576278985412157678985589876751214589872675983233123103742208899345951210026862001215954399880224730132136568526656881056991336986886896331996501886566258656311365685266568810569913369868868963319965018865662586563113656852665688105699133698688689633199650188656625865631136568526656881056991336986886896331996501886566258656311365685266568810569913369868868963319965018865662586563115997773553851169652456786055068765425696115835537779951111445658868843773685651365335631565863773488688565441111114456588688437736856513653356315658637734886885654411111144565886884377368565136533563156586377348868856544111461899988385443673048991500000000519984037634458388999816446189998838544367304899150000000051998403763445838899981641234678867875586685205942652898969898256249502586685578768876432112346788678755866852059426528989698982562495025866855787688764321123467886787558668520594265289896989825624950258668557876887643211234678867875586685205942652898969898256249502586685578768876432112346788678755866852059426528989698982562495025866855787688764321123662122673563335414654655879877212127789785564564145333653762212663211236621226735633354146546558798772121277897855645641453336537622126632112366212267356333541465465587987721212778978556456414533365376221266321895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598Solved all 12 digit numbersNumberIterationsDigitsResultant Palindrome100,900,509,906100,000,055,859104,000,146,950180,005,998,298300,000,185,539100,001,987,76512012112412914214358585865666646189998838544367304899150000000051998403763445838899981644618999883854436730489915000000005199840376344583889998164461899988385443673048991500000000519984037634458388999816412346788678755866852059426528989698982562495025866855787688764321895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598Solved all 13 digit numbersNumberIterationsDigitsResultant Palindrome1,000,007,614,6411,000,043,902,3201,000,006,653,7461,000,005,469,5484,000,096,953,6591,332,003,929,9951,000,201,995,6626,000,008,476,3791,200,004,031,6981,631,002,019,9931,000,006,412,2061,090,604,591,9301,600,005,969,1901251261271281391401411831841851861871886565656866666687878787878768236478976724413455368469647845654874696486355431442767987463286682364789767244134553684696478456548746964863554314427679874632861234678867875586685205942652898969898256249502586685578768876432146476268994355205755566889613764755746731698866555750255349986267464895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598895549975467412444422685224544649946445422586224444214764579945598159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951Solved all 14 digit numbersNumberIterationsDigitsResultant Palindrome10,090,899,969,90140,000,004,480,27914,104,229,999,9951381811827087872319937995996732122112578857767863993687677588752112212376995997399132159788389444247969944047982661126897487188000881784798621166289740449969742444983887951159788389444247969944047982661126897487188000881784798621166289740449969742444983887951Solved all 15 digit numbersNumberIterationsDigitsResultant Palindrome100,000,109,584,608100,000,098,743,648100,004,789,906,151100,079,239,995,161100,389,619,999,030200,000,729,975,309107,045,067,996,994105,420,999,199,982101,000,269,830,970104,000,047,066,970700,000,001,839,569100,000,050,469,737101,000,789,812,993100,907,098,999,571100,017,449,991,820890,000,023,937,399100,009,989,989,199101,507,024,989,944107,405,139,999,943100,057,569,996,821103,500,369,729,970900,000,076,152,049100,000,439,071,028120,000,046,510,993103,000,015,331,997100,617,081,999,573100,009,029,910,821107,000,020,928,910100,000,090,745,299102,000,149,322,944130,000,074,931,591100,120,849,299,2601501511521531541551561571581591631641651661671691701711721731741781791801891901911921981992002018073807373777783778378787777778076768282808787878787878792929292147517821017820137766258531124522433354114533342254211358526677310287101287157413874338821379754365139707922389885874785889832297079315634579731288334783147517821017820137766258531124522433354114533342254211358526677310287101287157413874338821379754365139707922389885874785889832297079315634579731288334783387433882137975436513970792238988587478588983229707931563457973128833478358432004534377668812885984239766675854845857666793248958821886677343540023485584320045343776688128859842397666758548458576667932489588218866773435400234856636465397223988868589438723468747648413686314846747864327834985868889322793564636658432004534377668812885984239766675854845857666793248958821886677343540023485663646539722398886858943872346874764841368631484674786432783498586888932279356463661774549867359976752553569855597325952466425952379555896535525767995376894547711774549867359976752553569855597325952466425952379555896535525767995376894547711595123986638798256345695585543997458545854799345585596543652897836689321595115951239866387982563456955855439974585458547993455855965436528978366893215951159512398663879825634569558554399745854585479934558559654365289783668932159513676445643312326889735963556943878639865568936878349655369537988623213346544676316852798979877566787844774463258467761167764852364477448787665778979897258611685279897987756678784477446325846776116776485236447744878766577897989725861497366346787871024259785276238898388208866880288389883267258795242017878764366379449736634678787102425978527623889838820886688028838988326725879524201787876436637943676445643312326889735963556943878639865568936878349655369537988623213346544676315978838944424796994404798266112689748718800088178479862116628974044996974244498388795115978838944424796994404798266112689748718800088178479862116628974044996974244498388795115978838944424796994404798266112689748718800088178479862116628974044996974244498388795112981651870269618822891011387448633891552774772551983368447831101982288169620781561892112981651870269618822891011387448633891552774772551983368447831101982288169620781561892112981651870269618822891011387448633891552774772551983368447831101982288169620781561892112981651870269618822891011387448633891552774772551983368447831101982288169620781561892116616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661166167237958848524555985641014556984266244449779444426624896554101465895554258488597327616611661672379588485245559856410145569842662444497794444266248965541014658955542584885973276166116616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661Solved all 16 digit numbersNumberIterationsDigitsResultant Palindrome6,000,000,039,361,4791,421,000,069,679,9961,000,650,998,992,3111,000,002,899,436,4011,000,100,396,492,2001,400,000,027,672,4987,090,000,039,309,9191,000,050,048,994,9571,003,000,024,749,9231,000,803,019,495,7111,030,020,097,997,900161162168175176177193194195196197787876787887929292929217745498673599767525535698555973259524664259523795558965355257679953768945477117745498673599767525535698555973259524664259523795558965355257679953768945477116852798979877566787844774463258467761167764852364477448787665778979897258611696356464454699645877422176000004533488433540000067122477854699645446465369611696356464454699645877422176000004533488433540000067122477854699645446465369611597883894442479699440479826611268974871880008817847986211662897404499697424449838879511661672379588485245559856410145569842662444497794444266248965541014658955542584885973276166116616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661166167237958848524555985641014556984266244449779444426624896554101465895554258488597327616611661672379588485245559856410145569842662444497794444266248965541014658955542584885973276166116616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661Solved all 17 digit numbersNumberIterationsDigitsResultant Palindrome10,000,000,730,931,02720,005,000,862,599,81911,450,360,479,969,99410,009,000,275,899,56910,059,430,139,999,23412,179,702,595,999,99179,000,000,445,783,59910,000,000,767,846,79710,000,000,673,402,33610,000,000,525,586,20610,005,000,760,994,24910,030,503,899,969,52412,000,009,694,736,29110,442,000,392,399,960160206207208209210229230231232233234235236829090909090112112111111111111111111145457866527997663444653236877754894438899883449845777863235644436679972566875454179588487572056993774972876634989826632577659889567752366289894366782794773996502757848859779588487572056993774972876634989826632577659889567752366289894366782794773996502757848859779588487572056993774972876634989826632577659889567752366289894366782794773996502757848859779588487572056993774972876634989826632577659889567752366289894366782794773996502757848859779588487572056993774972876634989826632577659889567752366289894366782794773996502757848859745772672672651139732189239727964147556436686594297045255552540792495686634655741469727932981237931156276276277544577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975Solved all 18 digit numbersNumberIterationsDigitsResultant Palindrome195,030,047,999,791,993100,000,078,999,111,766100,710,000,333,399,973100,000,002,973,751,552100,000,277,999,334,202110,300,361,999,869,090300,000,000,128,545,799104,300,000,514,769,945100,700,000,509,609,622120,906,490,499,909,290900,040,000,881,499,569100,072,100,489,999,238121,506,542,999,979,993106,096,507,979,997,951100,980,800,839,699,830600,000,000,606,339,049170,500,000,303,619,99620220320420521121221321421521621821922022122222722810510590901071071011011011011031031031031031121128899566748965874358576784476344886526578234896641354445314669843287562568844367448767585347856984766599888899566748965874358576784476344886526578234896641354445314669843287562568844367448767585347856984766599887958848757205699377497287663498982663257765988956775236628989436678279477399650275784885977958848757205699377497287663498982663257765988956775236628989436678279477399650275784885971662545996683346537994770004753335785026513598994469767964499895315620587533357400077499735643386699545266116625459966833465379947700047533357850265135989944697679644998953156205875333574000774997356433866995452661148846956853112695994986568587598546851289666436522256346669821586458957858656894995962113586596488411488469568531126959949865685875985468512896664365222563466698215864589578586568949959621135865964884114884695685311269599498656858759854685128966643652225634666982158645895785865689499596211358659648841148846956853112695994986568587598546851289666436522256346669821586458957858656894995962113586596488411596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951159656993445873017948889697545565364784201686389884648898368610248746356554579698884971037854439965695115965699344587301794888969754556536478420168638988464889836861024874635655457969888497103785443996569511596569934458730179488896975455653647842016863898846488983686102487463565545796988849710378544399656951159656993445873017948889697545565364784201686389884648898368610248746356554579698884971037854439965695145772672672651139732189239727964147556436686594297045255552540792495686634655741469727932981237931156276276277544577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754As of Thursday, November 13, 2008, solved 38.005% of all 19 digit numbersNumberIterationsDigitsResultant Palindrome1,000,600,400,194,191,2101,060,000,000,523,124,9951,000,000,079,994,144,3851,003,062,289,999,939,1421,186,060,307,891,929,990217226259260261101112119119119122226764220256667643474143237723258557664024982344432894204667558523277323414743467666520224676222214577267267265113973218923972796414755643668659429704525555254079249568663465574146972793298123793115627627627754445626658789764376224378489766538703888847836625984258559634369558524895266387488883078356679848734226734679878566265444456266587897643762243784897665387038888478366259842585596343695585248952663874888830783566798487342267346798785662654444562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544Iterations up to 261 for which no number was found223, 224, 225, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258.The intriguing part about these records is thatit takes an unreal amount of calculations to discover them,but it takes only a minute amount of calculations toprove that they terminate with a palindrome in the claimednumber of iterations.With a program, it would take less thanone second to prove that the above numbers do indeed solve out.I do not need a Guinness World Records Officialto be present during these calculations.You can take these numbers yourself,write up a quick program, and prove the results yourself.In fact, I have done so. Click on any of the numbers above, and youwill be shown a calculation page - computed in real-time (you can test this by entering your own unique number at the end of the URL) -proving that this number resolves into apalindrome in the displayed number of steps,showing you the calculations required to get there.Please contact me if you would like more information,or if you know that I have stated incorrect informationon this page. To the best of my knowledge, it is correctat this time.FRIDAY, OCTOBER 13, 2006 UPDATE:Vaughn Suite has completed the processing all 19 digit numbersusing a distributed network of 7 machines.He has found the following records:The 19 digit number 1,000,000,038,990,407,538 solves after 217 iterations, found on November 27, 2005.The 19 digit number 9,000,000,000,255,353,839 solves after 224 iterations, found on February 26, 2006.The 19 digit number 1,000,000,005,577,676,468 solves after 225 iterations, found on November 24, 2005.The 19 digit number 1,060,000,000,523,124,995 solves after 226 iterations, found on March 9, 2006.The 19 digit number 3,000,000,022,999,288,679 solves after 258 iterations, found on April 20, 2006.The 19 digit number 1,000,000,079,994,144,385 solves after 259 iterations, found on November 28, 2005.The 19 digit number 1,003,062,289,999,939,142 solves after 260 iterations, found on March 19, 2006.The 19 digit number 1,186,060,307,891,929,990 solves after 261 iterations, found on January 2, 2006,which is the same world record that my program already solved on November 30, 2005.No longer delays were found.Note: You may notice that I have previously stated,in my November 30, 2005 update,that Vaughn had already foundthe 259 iteration record,1,000,000,079,994,144,385,on July 26, 2005.When Vaughn was processing the 18 digit set on his work desktop last year,he started processing the 19 digit set on his laptop.This processing discovered this 259 iteration record.When he distributed the work into 15 subsets,he had to restart the 19 digit set from scratch,to be able to join it together properly.This processing rediscovered this result on November 28, 2005.RETIREMENT:Incidentally, at this time, I have only processed 9.782% of the 19 digit set.Why so little?My program has been running only part time, about one third of the time, on one machine.This lack of computer time is due to my retirement of this quest almost a year ago.I will eventually analyze my logs to find out when I stopped processing 24/7,and see how much CPU time I used since then until now.I should note my source code is available for anyserious requestto make use of it.I retired for several reasons.One is that Vaughn is using his own implementationof the same exponentially faster algorithm that I pioneered,which means his code is as fast as mine,all else being equal.Also, Vaughn improved the speed of the reverse and add procedureusing his own assembly code (optimized for the Pentium III, Pentium 4 and Athlon XP processors).And, he did so specifically for such short iteration spans,unlike the 196 Palindrome Quest,which cannot benefit from cache as greatly,since it processes numbers much larger than available CPU cache.My reverse and add code is from Benjamin Despres,which was optimized for the 196 Palindrome Quest.As you can see fromWade's Software Comparison Page,Vaughn's code is the fastest for smaller numbers.However, even for the 196 Palindrome Quest,Vaughn's assembly code istwice as fast as Ben's.To Ben's credit, Ben created his code a long time ago, in mid 2002,before he was able to make use of some of the newer CPU instructions.In 2002, Ben's had the fastest known application,responsible for taking the 196 Palindrome Quest from29 million digits to 45 million digits.Also, the largest factor in Vaughn's effort was that his application was distributedover seven machines. I only had one. I had access to seven CPUs,but they hadmore important thingsto spend their time on.Thus, I simply could not keep up with his much faster applicationon his incredible amount of CPU power he has available.It was a great effort, and I look forward to his analysis of the 20 digit set.SUNDAY, FEBRUARY 3, 2008 UPDATE:Fifteen months after completing processing the 19 digit numbers,Vaughn Suite has completed processing all 20 digit numbers bydistributing the work into 15 subsets and using the Pentium 4 HTand 2 Athlon XP machines from the 19 digit processing and alsothree more powerful computers: an Athlon 64, an Athlon 64 X2,and a Core 2 Quad machine. The subsets were executed simultaneouslyon different machines, or in multiple threads on the machines withhyperthreading or multi-core capabilities(Pentium 4 HT, Athlon X2, Core-2 Quad).Of the 5,808,378,560,022 numbers checked, there were5,459,760,062,742 lychrels (94.00%), while 348,618,497,280 numbersresolved into palindromes.Cumulative processing time on all the machines was 131,146,531 seconds = 1517.9 days (4 years and 2 months), but processing startedJanuary 6, 2007 and finished January 22, 2008.There are 58,083,785,600,220 (10 times as many) 21 digit numbers tobe checked, so that processing of that entire set of numbers willtake much longer with the current machines.Vaughn reports that the Core-2 machine was most responsible forthe quick completion since it runs optimized reverse and addsoftware 1.7 times as fast as the next fastest processor (Athlon X2),twice as fast as the Athlon 64, 2.5 times as fast as the Athlon XPsand almost 6 times as fast as each Pentium 4 thread in Hyperthreading mode.The new records are for 223, 253, 254, 255, 256 & 257 iterations:The 20 digit number 10,000,000,039,513,841,287 solves after 223 iterations, found on September 20, 2007.The 20 digit number 70,000,000,000,507,277,299 solves after 253 iterations, found on September 13, 2007.The 20 digit number 10,200,000,000,708,183,947 solves after 254 iterations, found on March 28, 2007.The 20 digit number 10,022,000,904,998,799,523 solves after 255 iterations, found on December 3, 2007.The 20 digit number 10,000,000,039,395,795,416 solves after 256 iterations, found on September 20, 2007.The 20 digit number 10,200,000,000,065,287,900 solves after 257 iterations, found on March 28, 2007.No longer iteration delays were found, and no iterations between237 iterations and 252 were found. The following iteration recordshave not yet been discovered:237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252.Palindrome Integer SequencesThe On-Line Encyclopedia of Integer Sequences,maintained byNeil J. A. Sloane,has quite a fewinteger sequences regarding palindromic numbers.My world records page is mentioned in the following:A006960 - The Reverse and Add! sequence starting with 196.A023108 - Positive integers which apparently never result in a palindrome under repeated applications of the function f(x) = x + (x with digits reversed).A023109 - Smallest number which requires exactly n iterations of Reverse and Add to reach a palindrome.A033665 - Number of 'Reverse and Add' steps needed to reach a palindrome, or -1 if never reaches a palindrome.A033670 - Reverse and Add! trajectory of 89.A065198 - n sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome starting with n.A065199 - Records for the number of 'Reverse and Add' steps needed to reach a palindrome.The following three were specifically created from my Most Delayed Palindromic Number World Records:A072216 - Consider the Reverse and Add! problem(cf. A001127);of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives values of N.A072217 - Consider the Reverse and Add! problem(cf. A001127);of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives number of steps N takes to converge.A072218 - Consider the Reverse and Add! problem(cf. A001127);of all the n-digit numbers N which eventually reach a palindrome, pick that number N which takes the greatest number of steps to converge (in case of a tie, pick the smallest N); sequence gives palindrome that is reached.ThisOn-Line Encyclopedia of Integer Sequenceshas over 100,000 sequences to date, and is well worth a look if you are interested in mathematical patterns. Back to Main Resume Page [Jason Doucette]:Traditional Resume / Curriculum Vitae —Projects / Games —Real Time Computer Graphics —Artificial Intelligence —World Records / 196 Palindrome Quest —Wallpapers / Desktops / Backgrounds —University Transcripts —Programming Windows, Fifth Edition, Errata Addendum—[Family]:Matthew Doucette —Wayne Doucette —Genesta Doucette—[Other Links]:Xona.com™ —Domain Hacks Suggest —Gibson Research Corporation —SpinRite 6.0 —Security Now! —Charles Petzold144,027visitors since August 18, 19991,640,377total page views since May 13, 1999Jason Allen Doucette / Xona.com™ |
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