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 reply to computer solution by Charlie)

196 is the first in a sequence of numbers for which it is unknown if the number ever reaches a palindrome.

Of all the numbers less than 10,000, the longest sequence is 24 for 89/98.  There are 249 that are unknown to stop or continue indefinitely, the smallest is the previously mentioned 196.

196 has been carried out for over 240,000,000 iterations, at which time it reached a 100 million digit number.

Of the numbers which do reach a palindrome, the 17 digit number 10,422,000,392,399,960 is currently the one which takes the longest to reach a palindrome in 236 iterations. The palindrome it forms is:
579922447678688885254995975774499665520124525649777656969656777946525421025566994477579599452588886876744229975

 

Posted by Brian Smith on 2004-04-05 09:21:29