Making Numbers into Palindromic NumbersDate: 10/11/95 at 15:20:15 From: Anonymous Subject: palindromes I'm interested in palindromic numbers. Apparently almost any number can be made into a palindromic number by reversing the digits and adding and then repeating the steps until you get a palindromic number. Some require many steps. I'm wondering if there is a list anywhere that tells which numbers can be made into palindromic sums and how many steps would be required. I'm interested in getting a list for numbers <1000 although I'd be happy to get a list for those <100. Do you know where I might get such a list? Date: 11/29/95 at 12:20:10 From: Doctor Ken Subject: Re: palindromes Hello! As far as I know, your question is an unanswered one in the general case, i.e. people don't know whether it's true for all numbers or not. But I think people have checked the first few cases, and they all work. I wrote a little program using the language PERL that checks all the numbers up to 200. Here's what it told me: 1 gets palindromic after 0 steps: 1 2 gets palindromic after 0 steps: 2 3 gets palindromic after 0 steps: 3 4 gets palindromic after 0 steps: 4 5 gets palindromic after 0 steps: 5 6 gets palindromic after 0 steps: 6 7 gets palindromic after 0 steps: 7 8 gets palindromic after 0 steps: 8 9 gets palindromic after 0 steps: 9 10 gets palindromic after 1 steps: 11 11 gets palindromic after 0 steps: 11 12 gets palindromic after 1 steps: 33 13 gets palindromic after 1 steps: 44 14 gets palindromic after 1 steps: 55 15 gets palindromic after 1 steps: 66 16 gets palindromic after 1 steps: 77 17 gets palindromic after 1 steps: 88 18 gets palindromic after 1 steps: 99 19 gets palindromic after 2 steps: 121 20 gets palindromic after 1 steps: 22 21 gets palindromic after 1 steps: 33 22 gets palindromic after 0 steps: 22 23 gets palindromic after 1 steps: 55 24 gets palindromic after 1 steps: 66 25 gets palindromic after 1 steps: 77 26 gets palindromic after 1 steps: 88 27 gets palindromic after 1 steps: 99 28 gets palindromic after 2 steps: 121 29 gets palindromic after 1 steps: 121 30 gets palindromic after 1 steps: 33 31 gets palindromic after 1 steps: 44 32 gets palindromic after 1 steps: 55 33 gets palindromic after 0 steps: 33 34 gets palindromic after 1 steps: 77 35 gets palindromic after 1 steps: 88 36 gets palindromic after 1 steps: 99 37 gets palindromic after 2 steps: 121 38 gets palindromic after 1 steps: 121 39 gets palindromic after 2 steps: 363 40 gets palindromic after 1 steps: 44 41 gets palindromic after 1 steps: 55 42 gets palindromic after 1 steps: 66 43 gets palindromic after 1 steps: 77 44 gets palindromic after 0 steps: 44 45 gets palindromic after 1 steps: 99 46 gets palindromic after 2 steps: 121 47 gets palindromic after 1 steps: 121 48 gets palindromic after 2 steps: 363 49 gets palindromic after 2 steps: 484 50 gets palindromic after 1 steps: 55 51 gets palindromic after 1 steps: 66 52 gets palindromic after 1 steps: 77 53 gets palindromic after 1 steps: 88 54 gets palindromic after 1 steps: 99 55 gets palindromic after 0 steps: 55 56 gets palindromic after 1 steps: 121 57 gets palindromic after 2 steps: 363 58 gets palindromic after 2 steps: 484 59 gets palindromic after 3 steps: 1111 60 gets palindromic after 1 steps: 66 61 gets palindromic after 1 steps: 77 62 gets palindromic after 1 steps: 88 63 gets palindromic after 1 steps: 99 64 gets palindromic after 2 steps: 121 65 gets palindromic after 1 steps: 121 66 gets palindromic after 0 steps: 66 67 gets palindromic after 2 steps: 484 68 gets palindromic after 3 steps: 1111 69 gets palindromic after 4 steps: 4884 70 gets palindromic after 1 steps: 77 71 gets palindromic after 1 steps: 88 72 gets palindromic after 1 steps: 99 73 gets palindromic after 2 steps: 121 74 gets palindromic after 1 steps: 121 75 gets palindromic after 2 steps: 363 76 gets palindromic after 2 steps: 484 77 gets palindromic after 0 steps: 77 78 gets palindromic after 4 steps: 4884 79 gets palindromic after 6 steps: 44044 80 gets palindromic after 1 steps: 88 81 gets palindromic after 1 steps: 99 82 gets palindromic after 2 steps: 121 83 gets palindromic after 1 steps: 121 84 gets palindromic after 2 steps: 363 85 gets palindromic after 2 steps: 484 86 gets palindromic after 3 steps: 1111 87 gets palindromic after 4 steps: 4884 88 gets palindromic after 0 steps: 88 89 gets palindromic after 24 steps: 8813200023188 90 gets palindromic after 1 steps: 99 91 gets palindromic after 2 steps: 121 92 gets palindromic after 1 steps: 121 93 gets palindromic after 2 steps: 363 94 gets palindromic after 2 steps: 484 95 gets palindromic after 3 steps: 1111 96 gets palindromic after 4 steps: 4884 97 gets palindromic after 6 steps: 44044 98 gets palindromic after 24 steps: 8813200023188 99 gets palindromic after 0 steps: 99 100 gets palindromic after 1 steps: 101 101 gets palindromic after 0 steps: 101 102 gets palindromic after 1 steps: 303 103 gets palindromic after 1 steps: 404 104 gets palindromic after 1 steps: 505 105 gets palindromic after 1 steps: 606 106 gets palindromic after 1 steps: 707 106 gets palindromic after 1 steps: 707 107 gets palindromic after 1 steps: 808 108 gets palindromic after 1 steps: 909 109 gets palindromic after 2 steps: 1111 110 gets palindromic after 1 steps: 121 111 gets palindromic after 0 steps: 111 112 gets palindromic after 1 steps: 323 113 gets palindromic after 1 steps: 424 114 gets palindromic after 1 steps: 525 115 gets palindromic after 1 steps: 626 116 gets palindromic after 1 steps: 727 117 gets palindromic after 1 steps: 828 118 gets palindromic after 1 steps: 929 119 gets palindromic after 2 steps: 1331 120 gets palindromic after 1 steps: 141 121 gets palindromic after 0 steps: 121 122 gets palindromic after 1 steps: 343 123 gets palindromic after 1 steps: 444 124 gets palindromic after 1 steps: 545 125 gets palindromic after 1 steps: 646 126 gets palindromic after 1 steps: 747 127 gets palindromic after 1 steps: 848 128 gets palindromic after 1 steps: 949 129 gets palindromic after 2 steps: 1551 130 gets palindromic after 1 steps: 161 131 gets palindromic after 0 steps: 131 132 gets palindromic after 1 steps: 363 133 gets palindromic after 1 steps: 464 134 gets palindromic after 1 steps: 565 135 gets palindromic after 1 steps: 666 136 gets palindromic after 1 steps: 767 137 gets palindromic after 1 steps: 868 138 gets palindromic after 1 steps: 969 139 gets palindromic after 2 steps: 1771 140 gets palindromic after 1 steps: 181 141 gets palindromic after 0 steps: 141 142 gets palindromic after 1 steps: 383 143 gets palindromic after 1 steps: 484 144 gets palindromic after 1 steps: 585 145 gets palindromic after 1 steps: 686 146 gets palindromic after 1 steps: 787 147 gets palindromic after 1 steps: 888 148 gets palindromic after 1 steps: 989 149 gets palindromic after 2 steps: 1991 150 gets palindromic after 2 steps: 303 151 gets palindromic after 0 steps: 151 152 gets palindromic after 2 steps: 707 153 gets palindromic after 2 steps: 909 154 gets palindromic after 2 steps: 1111 155 gets palindromic after 3 steps: 4444 156 gets palindromic after 3 steps: 6666 157 gets palindromic after 3 steps: 8888 158 gets palindromic after 3 steps: 11011 159 gets palindromic after 2 steps: 1221 160 gets palindromic after 2 steps: 343 161 gets palindromic after 0 steps: 161 162 gets palindromic after 2 steps: 747 163 gets palindromic after 2 steps: 949 164 gets palindromic after 3 steps: 2662 165 gets palindromic after 3 steps: 4884 166 gets palindromic after 5 steps: 45254 167 gets palindromic after 11 steps: 88555588 168 gets palindromic after 3 steps: 13431 169 gets palindromic after 2 steps: 1441 170 gets palindromic after 2 steps: 383 171 gets palindromic after 0 steps: 171 172 gets palindromic after 2 steps: 787 173 gets palindromic after 2 steps: 989 174 gets palindromic after 4 steps: 5115 175 gets palindromic after 4 steps: 9559 176 gets palindromic after 5 steps: 44044 177 gets palindromic after 15 steps: 8836886388 178 gets palindromic after 3 steps: 15851 179 gets palindromic after 2 steps: 1661 180 gets palindromic after 3 steps: 747 181 gets palindromic after 0 steps: 181 182 gets palindromic after 6 steps: 45254 183 gets palindromic after 4 steps: 13431 184 gets palindromic after 3 steps: 2552 185 gets palindromic after 3 steps: 4774 185 gets palindromic after 3 steps: 4774 186 gets palindromic after 3 steps: 6996 187 gets palindromic after 23 steps: 8813200023188 188 gets palindromic after 7 steps: 233332 189 gets palindromic after 2 steps: 1881 190 gets palindromic after 7 steps: 45254 191 gets palindromic after 0 steps: 191 192 gets palindromic after 4 steps: 6996 193 gets palindromic after 8 steps: 233332 194 gets palindromic after 3 steps: 2992 195 gets palindromic after 4 steps: 9339 197 gets palindromic after 7 steps: 881188 198 gets palindromic after 5 steps: 79497 199 gets palindromic after 3 steps: 3113 200 gets palindromic after 1 steps: 202 The only number that's missing from the list is 196. I performed the recursive flip-and-add process 10,000 times, and still 196 failed to turn into a palindrome. Pretty weird. I have a feeling that somewhere, somebody has found that 196 will yield eventually, but I still don't know. The next number that's missing is 295. Hope this is interesting to you! -Doctor Ken, The Geometry Forum And then, this came in.... From: Hoey Date: Thu, 12 Dec 96 I saw your articles in the Dr. Math archives about repeadedly adding a number to its reverse until a palindrome appears http://mathforum.org/dr.math/problems/barnes10.11.html . As you mentioned, it's unsolved. I've seen it in several popular math books and it is a perennial on rec.puzzles and sci.math; I imagine you will see it again on Dr. Math. You mentioned having tested it for 10,000 steps, but it's been taken much further. A report at http://www.fourmilab.ch/documents/threeyears.doc documents a search to over 2.4 million steps, or one million digits. The rec.puzzles archive (ftp://rtfm.mit.edu/pub/usenet/news.answers/puzzles/archive ) claims that the sequence has been tested for 9.48 million steps, or 3.9 million digits. You suggested that 196 will yield eventually. If you mean it will become a palindrome, I don't think that is very likely. The reason is that the number will only yield a palindrome if there is no carry out of any digit when the addition is performed. If the digits are randomly distributed, the probability of this is about 2^(-n/2) for an n-digit number. The number of digits is about 2/5 of the number of steps taken, and will certainly increase at least once every five steps. So the probability that the number will eventually reach a palindrome on the k'th step should be less than 2^(-k/10), and the sum of that for k=3000000 to infinity is something like 10^-90000. There are some numbers in other bases for which the process can be shown not to terminate (http://mathpages.com/home/kmath004.htm , for example) but that is because a particularly orderly kind of progression is followed. None is now known in base 10, though I wouldn't be surprised if one is found. But I doubt that 196 will turn out this way either. It looks to me like it will probably continue to grow chaotically, and I don't think our mathematics is up to proving anything about it. -Dan Hoey Additional Note from Dr. Ken: Many thanks to Dan! It hadn't occurred to me to approach the problem probabilistically. As for whether mathematics will be able to shed more light on whether 196 and the other integers eventually yield to the palindrome-generation process, I recognize the difficulty of the problem, but I'm a little more optimistic. After all, we humans have devised a method for computing any digit of Pi directly, without having to first compute the previous digits, and I think that's pretty impressive. See http://pi.lacim.uqam.ca/eng/ for more details about that. -Dr. Ken Date: 12/05/2001 at 23:33:50 From: Wade VanLandingham Subject: 196 palindromic solution I notice that Dr Ken commented that 196 had been reversed over 10,000 times, with no solution. Then Dan Hoey commented that it had been taken to over a million digits. I would like to inform you that 196 has been taken to over 56 million iterations, with a result over 23,500,000 digits long, and there is still no palindrome... Details can be found at http://www.geocities.com/jwvanno/ Thank you for your wonderful site. I refer to it REGULARLY. Best wishes Wade VanLandingham |
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