A palindrome is a word which reads the same backwards as it does forwards. Similarly, a number having this quality is considered 'palindromic'.

Consider the number 23. Reverse and add that number to 23 to yield the palindrome 55. This has taken one step.

Now consider the same reversal-summation process for 37. Over 2 steps it becomes the palindrome 121(37,73/110,011/121).

QUESTION: What 2 digit number requires the most stages to become a palindrome? How many stages is that, and what is the palindrome?


NOTES, THOUGHTS, CONSIDERATIONS:
1] It will become quickly evident that certain combinations will run out fast (like, why use 32 when 23 is its reverse).

2] Spreadsheet calculator - If you need one. Copy the formula ' =A1+A2 ' into cell 3 of a column. Copy it into each successively descending odd cell as may be needed. Format the column as 'Number' having no decimal places so as to override that particular time when the Exp. Not. default would normally cut in.

Enter your chosen number into A1 and continue to place the reverse total of each odd row into the cell of the next even row.

3] Obviously such sequences do occur beyond 2 digit numbers and while I have some 3 digit challenges in my personal literature, I am wondering if there is some 'easy' way of when a number becomes a palindrome - obviously I assume that it will happen with any number eventually (but should I pose that question, or even present some of the 3 digit challenges?).

In the 3-digit range, other than the ones that exceed the capacity of the language, the maximum steps is 23:

 1  968
 2  1837
 3  9218
 4  17347
 5  91718
 6  173437
 7  907808
 8  1716517
 9  8872688
 10  17735476
 11  85189247
 12  159487405
 13  664272356
 14  1317544822
 15  3602001953
 16  7193004016
 17  13297007933
 18  47267087164
 19  93445163438
 20  176881317877
 21  955594506548
 22  1801200002107
 23  8813200023188

These same steps apply for 187, 286, 385, 583, 682, 781 and 880.

The 3-digit numbers that exceed the capacity are 196, 295, 385, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986.

Posted by Charlie on 2004-04-05 11:02:36