Andy

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?

Thanks.

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.