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Lychrel Numbers

Description

The Lychrel number algorithm is defined as follows: Starting with a positive integer perform the iterative process of summing a number with its reversed number until you get to a palindrome number (identical to its reversed number).

For example starting with 19 you get the series
\(a_0(19)=19\)
\(a_1(19)=19+91=110\)
\(a_2(19)=110+11=121\), which is palindrome.

Lychrel numbers are defined as numbers, that no matter how often you iterate this process never form a palindrome. Finding a Lychrel number is an unsolved problem.

Note the obvious: The series of numbers you generate are monotonically increasing, as reversing a positive integer never gives a negative number. So you cannot get into a cycle of repeating values to proof a Lychrel number by an example going into a cycle of all non-palindrome numbers.

Typically the process does not take many iterations to stop at a palindrome. The idea to add the reversed number is to create symmetry, which causes a palindrome, if there is no carryover, for example as in 18+81=99.

How many of all positive integers \(n<100000\) don't turn to a palindrome within the first 100 iteration steps?

Note: For this problem I define the first iteration step as computing \(a_1(n)\) from \(a_0(n)=n\), not setting \(a_0(n)\) to \(n\), which is just defining the seed value of the process. That means numbers being a palindrome themselves take no step and end at their beginning.

 


Question

How many of all positive integers \(n<100000\) don't turn to a palindrome within the first 100 iteration steps of the Lychrel number algorithm?


How many of all positive integers \(n<100000\) don't turn to a palindrome within the first 100 iteration steps of the Lychrel number algorithm?