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 re(2): solutions by Krishneel Nair)

Three points:

1.  Apology!  I misused the checksum; I overlooked the fact that the checksum should have taken into account all of the addends - in fact, had I been thinking clearly at the time, I would have realised that the concept was inappropriate in the way I proposed it.  In fact, the checksum of all of the addends would have equaled the chacksum of the palidrome (however such verification is not necessarily conclusive as one error may 'validate' another).

2.  The checksum of 23 is indeed 5; your direction to this sample did cause my rethink, thankyou for providing a reassessment.

3.  A lead into the total solution has been posted in earlier comments

 

Posted by brianjn on 2004-04-27 02:16:20