Palindromic Numbers

A palindrome is a word, phrase, or number that can be read the same way backwards or forwards. Some examples of Palindromic numbers are 11, 121, 666, and 45354.

There is a famous unsolved number problem called the "Palindrome Conjecture." This conjecture states that any positive integer can be made into a palindrome in a finite number of steps by following a simple procedure. In this procedure, reverse the digits of the original number and add the two numbers together. If the sum is a palindrome, the procedure ends; if not, the process is repeated until a palindrome is produced.

For example, 138 is not a palindrome, but it can produce a palindrome in one step. Take 138, reverse the digits to get 831, then add them together to get 138 + 831 = 969, a palindrome.

Similarly, is 168 a palindrome? No, so reverse the digits to get 861, then add them together to get 1029, still not a palindrome. So, repeat the process, reversing the digits and adding: 1029 + 9201 = 10230, still not a palindrome. So, repeat the process: 10230 + 03201 = 13431, a palindrome in three steps!

No one knows if this conjecture is true. Mathematicians have found numbers which have not produced palindromes, even after thousands of steps. One example is the number 196, which has been taken to hundreds of thousands of steps by computers without producing a palindrome.

Now use this process to record the number of steps and the palindrome produced for each of the numbers in the table below:

Number Number of Steps Palindrome

138 1 969

168 3 13431

68 _____

728 _

97

_

472 _

835 _

988 _

561

_

193 _

553 _

86 _

918

_

192 _

364 _

829 _

89

_________