Mathematical Investigation of Two Digit Palindromes

To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.

Ivars Peterson

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A number is called a palindromic number if it reads the same both forward and backward. In other words, it has reflectional symmetry across a vertical axis. Few palindromic number are 11, 121, 31413. English sentences can also be palindromic, like do geese see god or murder for a jar of red rum or step on no pets.

In this blogpost, we will consider a problem related to palindromes. For this, let’s pick an arbitrary number, reorder the digits in reverse and add it to the original number. It is said that repeating this operation eventually leads, at some point, to a palindrome. For example, suppose we pick 96. See the steps below to obtain the palindrome!

Step 1. Consider the number 96. Reverse of this number is 69. Their sum is 165.

Step 2. Now, we consider the number 165. Its reverse is 561. Their sum is 726.

Step 3. Now, we consider 726. Its reverse is 627. Their sum is 1353.

Step 4. Now, we consider 1353. Its reverse is 3531. Their sum is 4884.

Definitely, 4884 is a palindrome! Here, after 4 repetitions of the operation the number 96 arrived at the palindrome 4884.

Let’s try again with another number, 59.

Step 1. Consider the number 59. Reverse of this number is 95. Their sum is 154.

Step 2. Now, we consider the number 154. Its reverse is 451. Their sum is 605.

Step 3. Now, we consider 605. Its reverse is 506. Their sum is 1111, a palindrome!

Here, after 3 repetitions of the operation the number 59 arrived at the palindrome 1111.

The number 89 yield a palindrome after 24 iterations and we obtain 8813200023188 (a thirteen-digit palindrome!). Let’s see how.

• Step 1. 89 + 98 = 187
• Step 2. 187 + 781 = 968
• Step 3. 968 + 869 = 1837
• Step 4. 1837 + 7381 = 9218
• Step 5. 9218 + 8129 = 17347
• Step 6. 17347 + 74371 = 91718
• Step 7. 91718 + 81719 = 173437
• Step 8. 173437 + 734371 = 907808
• Step 9. 907808 + 808709 = 1716517
• Step 10. 1716517 + 7156171 = 8872688
• Step 11. 8872688 + 8862788 = 17735476
• Step 12. 17735476 + 67453771 = 85189247
• Step 13. 85189247 + 74298158 = 159487405
• Step 14. 159487405 + 504784951 = 664272356
• Step 15. 664272356 + 653272466 = 1317544822
• Step 16. 1317544822 + 2284457131= 3602001953
• Step 17. 3602001953 + 3591002063 = 7193004016
• Step 18. 7193004016 + 6104003917 = 13297007933
• Step 19. 13297007933 + 33970079231 = 47267087164
• Step 20. 47267087164 + 46178076274 = 93445163438
• Step 21. 93445163438 + 83436154439 = 176881317877
• Step 22. 176881317877 + 778713188671 = 955594506548
• Step 23. 955594506548 + 845605495559 = 1801200002107
• Step 24. 1801200002107 + 7012000021081 = 8813200023188

Will this process always result in a palindromic number? There are total 90 two-digit numbers, from 10 up to 99. It can be confirmed that starting from any two-digit number from 10 to 99 leads to a palindrome.

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