My friend Justin, said that when his children were in 6th grade, they were taught that if you take any number, reverse it and add the two together, a palindrome will result. If not, continue performing the above operations, and a palindrome will eventually result.

 eg:  10         23         78     165     726     1353
     +01        +32        +87  / +561  / +627  / +3531
     ---        ---        --- /   --- /  ---- /   ----
      11         55        165     726    1353     4884
We've both gone all the way through 0 to 195, and the above rule held true. But we can not get a result for 196. He wrote a program in FORTRAN that kept running through and produced sums over 63000 digits long, still with no result. I made a program in BASIC, and got a sum over 100 digits long and I stopped it and gave up.

So my question is: Is there a simple formula to determine what the final palindromatic sum will be, and/or is there a formula to determine how many rotations and additions are necessary to obtain a palindrome?


Hi Andy

This is a facinating question. We wrote a program in Mathematica, as you did in BASIC, and with 196 got to a sum with 500 digits without seeing a palindrome.

We were able to find two references to this problem:

Heiko Harborth, On Palindromes, Mathematics Magazine, (1973) 96-99 and
C.W. Trigg, Palindromes in addition, Mathematics Magazine, 40(1967) 26-28.

In the 1973 paper Harborth says that Trigg, in 1967, checked all integers less than 10,000 and found 249 of them do not seem to have a palindrome in the sequence of sums that you produce. 196 is undoubtedly one of the 249 numbers he identified. Harborth then goes on to solve a somewhat modified problem. We contacted Heiko Harborth and he says that the situation has not changed since 1973.

So the answer to your question is that the the grade 6 teacher was unduely optimistic.

In number theory there are many examples of conjectures like this that are easily stated but very difficult to prove or disprove. The most famous is Fermat's Last Theorem.

Thanks for a great question.

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