  Recently, Cilleruelo, Luca, & Baxter proved, for all bases b >= 5, that every
natural number is the sum of at most 3 natural numbers whose base-b
representation is a palindrome. However, the cases b = 2, 3, 4 were left
unresolved.
  We prove, using a decision procedure based on automata, that every natural
number is the sum of at most 4 natural numbers whose base-2 representation is a
palindrome. Here the constant 4 is optimal. We obtain similar results for bases
3 and 4, thus completely resolving the problem.
  We consider some other variations on this problem, and prove similar results.
We argue that heavily case-based proofs are a good signal that a decision
procedure may help to automate the proof.
