Palindrome Created by Adding Reverse Numbers and Lychrel Numbers

A “palindrome” is a number that is the same even when reversed such as 121 or 99 or 14641 and even 5.

When any number is reversed and then added to its reverse, and this process is repeated, it will eventually become a palindrome. Unless, it does not, in which case we call these numbers Lychrel numbers, coined by Dutch mathematician, Wade Van Landingham after his girl friend Cheryl.

For example, if we take 15 plus 51 we get 66, a palindrome after one step (iteration). If we take 47 plus 74, we get 120 and if add 021 to it we get 141, a palindrome after two steps. If we take  the number 381, it takes 4 steps for it to become a palindrome: 13431.

I discovered the smallest Lychrel number by applying this process to the first 200 numbers when the number 196 did not work after 100 steps. I even checked my math thinking I made  mistake. Then I did some research and found that after 2,415,836 steps, the number had 1,000,000 digits and was not a palindrome. However, there is no proof that Lychrel Numbers exist. The smallest 26 possible Lychrel Numbers are:

196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997.

Don’t try these because you may get frustrated; or if you want to test them, try. If you get a palindrome, it is because you made a mistake along the way. Find your mistake.

I challenged the children to choose numbers that require a certain number of steps to reach a palindrome and then find that palindrome. For example, k/1st graders start with numbers that require 1 step (12 + 21= 33, so record the number 33). 2/3rd graders try numbers that require 3 steps. 4/5th graders try any number except for 196 which is a Lychrel number (will never become a palindrome).

The numbers in bold are considered challenging (4 steps or more to reach a palindrome).

After you are reaching palindromes in 2 or 3 steps, move on to 4 and 5 steps and challenge yourself as far as you wish to go.

The worksheet and the answer key for the first 200 numbers is attached as a pdf. 

About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers:

• 56 becomes palindromic after one iteration: 56+65 = 121.

• 57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.

• 59 becomes a palindrome after 3 iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111

• 89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188.

• 10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps.

• 1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which is the current world record for the Most Delayed Palindromic Number. It was solved by Jason Doucette's algorithm and program (using Benjamin Despres' reversal-addition code) on November 30, 2005.

• On January 23, 2017 a Russian schoolboy Andrey S. Shchebetov announced on his web site that he found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. This sequence started with 1,186,060,307,891,929,990 - by then the only publicly known number found by Jason Doucette back in 2005. On May 12, 2017 this sequence was extended to 108864 terms in total and included the first 108864 delayed palindromes with 261-step delay. The extended sequence ended with 1,999,291,987,030,606,810 - its largest and its final term.

• Any number from this sequence could be used as a primary base to construct higher order 261-step palindromes. For example, based on 1,999,291,987,030,606,810 the following number 199929198703060681000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001999291987030606810 also becomes a 238-digit palindrome 4562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 4562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 after 261 steps.

The number resulting from the reversal of the digits of a Lychrel number is also a Lychrel number.