This article is based on a recent Numberphile video.
Take any positive integer number.
Is the number a palindrome (does it read the same back-to-front)?
If not, write the current number backwards and add this to the current number.
This sum is the new number.
We keep track of how long it takes for a number to turn palindromic. As you can see from the example above, 39 took two steps. Some numbers (such as 22, 101, 666, 1221 …) are already palindromes, and take zero steps. By definition, all single digit numbers are also palindromes.
Most numbers turn into palindromes in just a couple of steps. e.g. 21 turns into a palindrome in one iteration. Even a larger number, such as 1967, turns into a palindrome with just five steps. Do all numbers turn into palindromes? If so, what numbers take the longest number of steps? What is the most delayed palindrome so far discovered?
All numbers less than 100, (with the exception of 89 and 98), take six or fewer iterations; most take less. It should be no surprise that 89 and 98 take the same number of steps as they are canonical forms of themselves (89 reversed is 98!). They both take 24 steps and the final palindromic number generated is 8813200023188.
Something interesting happens. 187 takes 23 steps (one less than 89; but that's not a surprise, because if you look at the sequence above for 89 you see that the second term is 187, and after that they, of course, follow the same chain).
The more interesting feature, however, is 196. It does not quickly form a palindrome. In fact, we don't know if it ever does!
Quite a lot of processing power has been dedicated to researching if 196 eventually becomes a palindrome. How much? Well, using distributed computing, Romain Dolbeau went through a billion iterations to produce a number with 413,930,770 digits, and still no palindrome was seen! (He went on to pass 600 million digits without finding anything).
Similar computerized brute force searches to find palindromes on these other Lychrel numbers have been attempted with no success. It's looking likely that these numbers never converge to palindromes, but this property has not yet been mathematically proven.
(The other potential point of contention between purists is if the palindrome test is performed before the first addition, or after. e.g. If before, obviously 9999 is not Lychrel, but if you require a least one iteration before test, it is, as it has never been found to palindrome after starting).
Interestingly 89 takes an unusually long time to palindrome. The 24 steps (rendered above), is the largest number of steps (of non-Lyrchel numbers) less than 10,000.
For numbers less than 10,000 a staggering 29.9% of all numbers turn palindromic in one (or less) iterations.
For numbers less than 10,000 over 80.5% turn palindromic in four or less iterations.
Obviously, if a number is Lychrel, so is its palindrome (unless you are a purist, in which case you simply take the smallest as a seed).
The word 'palindrome' comes from the Greek 'palíndromos', which means "running back again".
Below 1,000,000 the longest delayed palindrome requires 64 steps, and this is 150,296.
In addition to searching for (and proving, or disproving Lychrel numbers), researchers have used computers to find the longest delayed palindrome they can (that is not Lychrel). You can find their research here.
As the length of the starting number changes, so does the chance that a number will be Lychrel. By the time numbers are at 18 diigts of length, over 90% of them turn into Lychrel numbers and never convert to palindromes.
The longer a number gets, the smaller the chance that a palindrome will be generated (a palindrome can only be formed if there is no carry when two digits are added). Considering how large the potential Lychrel numbers have been tested to, even if not proved yet, it's incredibly unlikely they will ever converge to palindromes.
Sadly, other than interest, and an excuse to write code, I can think of no practical used for Lychrel numbers!