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A palindrome is usually a word (or sentence) which reads the same whether read forwards or backwards: for example "Madam, I'm Adam" (or her reply: "Eve").
We can also have palindromic numbers, such as 121, or 14641.
Here's a "recipe" for creating palindromic numbers:
Take any positive integer; reverse it (adding leading zeroes if necessary) and add this reversal to the original number. Repeat the operation using the result just obtained and eventually you will arrive at an answer which is palindromic.
Example (all numbers dozenal) :
| Start with | 39 | |
| reverse it | 93 | |
| add | 110 | |
| reverse | 011 | |
| add | 121 |
121 is palindromic.
| Start | 293 | |
| reverse | 392 | |
| add | 665 | |
| reverse | 566 | |
| add | 100E | |
| reverse | E001 | |
| add | 10010 | |
| reverse | 01001 | |
| add | 11011 |
11011 is palindromic
I've used base twelve, as usual, for my examples, but any base will do.
Now - is it always true? Will this method always produce a palindromic number from any starting number and in any base of numeration?
In bases four and five all numbers less than "1000" produce palindromes by this method. I haven't tried base six onwards yet.
If you like playing with numbers that should keep you busy for a while ...
Note: in base ten there are apparently thirteen numbers less than a thousand which do not produce a palindromic number even after thousands of steps. Will they eventually produce a palindrome? and how many steps will it take?
Anyone volunteering to find out which ones they are?