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The Mystery 196
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My last entry has reminded me a mystery that I encountered when I was a kid. It gravitates around the reverse-and-add procedure. I explain: take a number such as 934. If you reverse the digits you get 439. Adding the two numbers you get 1373. Doing the same reverse-and-add thing again with 1373 you get 1373 + 3731 = 5104. Another iteration gives 5104 + 4015 = 9119. We stop here because we have found what we were looking for: a palindromic number. In case you didn't know what is a palindromic number, that's a number which read the same from right to left as from left to right. Of course you can also say that some words are palindromic. The word "dad" is one of them.

So given a number, we consider the sequence of numbers obtained using this reverse-and-add method to compute the next one. We already have the following sequence: 934, 1379, 5104, 9119.

I have written a little script to compute the palindromic end of such sequences, taking every integer as starting point. The output of the script is below. It is to be read as follow: the first number is the starting point, the second number (between brackets) is the length of the sequence and the last number is the palindrome which has ended the sequence.

0 -> (0) 0
1 -> (0) 1
2 -> (0) 2
3 -> (0) 3
4 -> (0) 4
5 -> (0) 5
6 -> (0) 6
7 -> (0) 7
8 -> (0) 8
9 -> (0) 9
10 -> (1) 11
11 -> (0) 11
12 -> (1) 33
13 -> (1) 44
14 -> (1) 55
15 -> (1) 66
16 -> (1) 77
17 -> (1) 88
18 -> (1) 99
19 -> (2) 121
20 -> (1) 22
21 -> (1) 33
22 -> (0) 22
23 -> (1) 55
24 -> (1) 66
25 -> (1) 77
26 -> (1) 88
27 -> (1) 99
28 -> (2) 121
29 -> (1) 121
30 -> (1) 33
31 -> (1) 44
32 -> (1) 55
33 -> (0) 33
34 -> (1) 77
35 -> (1) 88
36 -> (1) 99
37 -> (2) 121
38 -> (1) 121
39 -> (2) 363
40 -> (1) 44
41 -> (1) 55
42 -> (1) 66
43 -> (1) 77
44 -> (0) 44
45 -> (1) 99
46 -> (2) 121
47 -> (1) 121
48 -> (2) 363
49 -> (2) 484
50 -> (1) 55
51 -> (1) 66
52 -> (1) 77
53 -> (1) 88
54 -> (1) 99
55 -> (0) 55
56 -> (1) 121
57 -> (2) 363
58 -> (2) 484
59 -> (3) 1111
60 -> (1) 66
61 -> (1) 77
62 -> (1) 88
63 -> (1) 99
64 -> (2) 121
65 -> (1) 121
66 -> (0) 66
67 -> (2) 484
68 -> (3) 1111
69 -> (4) 4884
70 -> (1) 77
71 -> (1) 88
72 -> (1) 99
73 -> (2) 121
74 -> (1) 121
75 -> (2) 363
76 -> (2) 484
77 -> (0) 77
78 -> (4) 4884
79 -> (6) 44044
80 -> (1) 88
81 -> (1) 99
82 -> (2) 121
83 -> (1) 121
84 -> (2) 363
85 -> (2) 484
86 -> (3) 1111
87 -> (4) 4884
88 -> (0) 88
89 -> (24) 8813200023188
90 -> (1) 99
91 -> (2) 121
92 -> (1) 121
93 -> (2) 363
94 -> (2) 484
95 -> (3) 1111
96 -> (4) 4884
97 -> (6) 44044
98 -> (24) 8813200023188
99 -> (0) 99
100 -> (1) 101
101 -> (0) 101
102 -> (1) 303
103 -> (1) 404
104 -> (1) 505
105 -> (1) 606
106 -> (1) 707
107 -> (1) 808
108 -> (1) 909
109 -> (2) 1111
110 -> (1) 121
111 -> (0) 111
112 -> (1) 323
113 -> (1) 424
114 -> (1) 525
115 -> (1) 626
116 -> (1) 727
117 -> (1) 828
118 -> (1) 929
119 -> (2) 1331
120 -> (1) 141
121 -> (0) 121
122 -> (1) 343
123 -> (1) 444
124 -> (1) 545
125 -> (1) 646
126 -> (1) 747
127 -> (1) 848
128 -> (1) 949
129 -> (2) 1551
130 -> (1) 161
131 -> (0) 131
132 -> (1) 363
133 -> (1) 464
134 -> (1) 565
135 -> (1) 666
136 -> (1) 767
137 -> (1) 868
138 -> (1) 969
139 -> (2) 1771
140 -> (1) 181
141 -> (0) 141
142 -> (1) 383
143 -> (1) 484
144 -> (1) 585
145 -> (1) 686
146 -> (1) 787
147 -> (1) 888
148 -> (1) 989
149 -> (2) 1991
150 -> (2) 303
151 -> (0) 151
152 -> (2) 707
153 -> (2) 909
154 -> (2) 1111
155 -> (3) 4444
156 -> (3) 6666
157 -> (3) 8888
158 -> (3) 11011
159 -> (2) 1221
160 -> (2) 343
161 -> (0) 161
162 -> (2) 747
163 -> (2) 949
164 -> (3) 2662
165 -> (3) 4884
166 -> (5) 45254
167 -> (11) 88555588
168 -> (3) 13431
169 -> (2) 1441
170 -> (2) 383
171 -> (0) 171
172 -> (2) 787
173 -> (2) 989
174 -> (4) 5115
175 -> (4) 9559
176 -> (5) 44044
177 -> (15) 8836886388
178 -> (3) 15851
179 -> (2) 1661
180 -> (3) 747
181 -> (0) 181
182 -> (6) 45254
183 -> (4) 13431
184 -> (3) 2552
185 -> (3) 4774
186 -> (3) 6996
187 -> (23) 8813200023188
188 -> (7) 233332
189 -> (2) 1881
190 -> (7) 45254
191 -> (0) 191
192 -> (4) 6996
193 -> (8) 233332
194 -> (3) 2992
195 -> (4) 9339
196 ->

Up to 9, numbers are one digit long and therefore are palindromic, so there was not much to see. Afterwards sequences are surprisingly short with the exception of 89 and 98 whose sequences are of length 24, 167 (length 11), 177 (length 15) and 187 (length 23). Incidentally I like the case of 28 and 29 which are such that 28 + 82 = 29 + 92. There are a couple of others like that (consecutive integers giving birth to the same palindromic number).

Anyway, I haven't given the results for 196. Well, that where the mystery is: nobody has ever found the palindromic number generated by 196. We (mathematicians) do not even know whether it exits or not. All we know is that some computers have run for months computing the sequence starting with 196, and nothing.

There are other numbers like that. To find some, it is enough to simply take the numbers which appear in the sequence generated by 196. The next number after 196 is 196+691=887. 887 exhibits the same strangeness (of course), but 196 is the smallest of them... As show on my script's output...

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