More on Palindromes by Reversal-Addition
Charles W. Trigg · Mathematics Magazine · 1972
When the digits of the integer N are written in reverse order, the integer N' is obtained. Let N + N' = Sl, SI + S' = S2, S2 + S2 = S3,..,Sk.l + Sk= Sk It has been conjectured that for every N there is a k for which Sk is a palindrome. The validity of this conjecture for N < 104 has been questioned in [1]. There it is reported that 249 integers < 104 have reversal-addition sequences that are palindrome-free up to at least k = 100. All one-digit and two-digit integers lead to palindromes. No other N < 1O4 requires more operations to produce a palindrome than the 24 required to produce S24(89) = 8813200023188. The search for palindrome-free sequences is now extended to N < 105. To exhaust the field of five-digit integers, only 3420 basic integers need to be dealt with. A basic integer is the smallest member and representative of a family all of which have the same S1. Thus S1(10321) = 22622 indicates that 10321, 11311, 12301, 20320, 21310, and 22300 each produces this palindrome in one operation. These family members have the same middle digits and their symmetrically located digit pairs have the same sums. Computation can be reduced further by recognizing that