Reverse and Add Rule for Finding Palindromes

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Given any numbers, you can use the following simple algorithm to find other palindromes.

Step 1: Start with any number. Call it original number. Reverse the digits of the original number

Step 2: Call the number whose digits are reversed new number. Add the new number to your original number.

Call the number found by adding the new number to the original number test number

Step 3: If test number is a palindrome, you are done. If not, use your test number as your original number and repeat the steps above

Example #1:

75          Reversing 75 gives 57  Adding 75 and 57 gives 132

Reversing 132 gives 231        Adding 132 and 231 gives 363

Example #2:

255        Reversing 255 gives 552     Adding 255 and 552 gives 807

Reversing 807 gives 708            Adding 807 and 708 gives 1515

Reversing 1515 gives 5151 Adding 1515 and 5151 gives 6666.

Now we are done! Now, here is your puzzle. Find the number of steps required to create palindromes from 2-digit numbers. For example, 77 is 0 steps; 23 and 32 are one step; and 19 and 91 are two steps. Find 4 numbers less than 100 that require exactly 4 additions to obtain palindromes. Once you solve this for all 2-digit numbers, try three digit numbers. There are some very interesting patterns in the numbers of steps. Also, there are a few numbers that have never been solved to create a palindrome like 196 or 691. Please don't try it as it has been tested to 200,000 steps. The solution for 89 is fascinating; if you are good at adding numbers, try it.

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