The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Mathematical Palindromes

Date: 01/17/97 at 19:36:43
From: Anonymous
Subject: Fwd: Mathematical palindromes

Do you know anything about patterns concerning mathematical 
palindromes and how they occur?  I am doing a math project for school.  
I am in the 4th grade.  I have discovered that if you add the reverse 
of a number to a number, you eventually will come to a palindrome.  
If they are 2-digit numbers, the sum of the 2 digits in the original 
number will determine the number of steps it takes to get to the 
palindrome.  I've also found information on the palindrome 121.  
Are there any other patterns or facts about palindromes?  I can't find 
much.  Thank you for your help.  


Date: 01/18/97 at 12:00:35
From: Doctor Lynn
Subject: Re: Fwd: Mathematical palindromes

Dear Samantha,

As far as I can tell, this is an unsolved problem of mathematics.  
There is no easy way to tell how many steps it will take, but I think 
you are right to think that with two digit numbers it is based 
approximately on the digit sum.

Two numbers which are interesting in this field are 89 and 196.
89 takes 24 steps before it reaches a palindrome (and so does its
reverse, 98), and needs a pretty powerful calculator to show the
final answer.  A computer program is best, but even then it
needs a mathematical programming language which can cope with
big numbers.  

196 doesn't seem to reach a palindrome.  To be more precise,
about ten years ago, (the last time I heard about this) no one knew
whether or not it did, so they will have tested it to thousands of
steps.  They may have found proof whether or not it does by now, but
I haven't heard about it.

With the number 121, have you noticed the pattern of squares of 
numbers made of rows of 1s?

These are:

1:              1
11:             121
111:            12321
1111:           1234321
11111:          123454321
111111:         12345654321

These are palindromes whose squares are palindromes too.

I'm very pleased you're showing an interest in mathematics and I'm 
sorry I couldn't help more.

If you are interested in number patterns, you might find the number 
142857 interesting. Try multiplying it by the numbers from 1 to 7...

-Doctor Lynn,  The Math Forum
 Check out our web site!   

Note from the archivist: see also   ,
which deals a little more extensively with the unsolved problem
mentioned above.
Associated Topics:
Elementary Number Sense/About Numbers
Middle School Number Sense/About Numbers

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.