Making Numbers into Palindromic Numbers

Date: 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