In a test I've found the following exercise:

We say $n \in \mathbb{N}$ is reflexive if is the sum of two naturals $x$ and $y$ such that $y$ has the same digits of $x$ witten in the inverse order (if $x=123$ then $y$ is $321$). Find all the reflexive natural numbers.

Now, I've tried to describe them using numbers of the form $10\dots 01$ but I assume there is a more elegant characterization of reflexive numbers.

Any ideas? Thank you!

  • $\begingroup$ Is $11=10+01$ reflexive? $\endgroup$ – Hagen von Eitzen Jun 13 '15 at 13:40
  • $\begingroup$ I've to be honest: by the text of exercise it is not very clear. Well, I think 11 is reflexive and so you can assume $n$ is reflexive in this kind of cases. $\endgroup$ – Sabino Di Trani Jun 13 '15 at 13:43
  • $\begingroup$ Let $x=\overline{a_1a_2\ldots a_m}$, $y=\overline{a_m a_{m-1}\ldots a_1}$. Then $$x+y=(10^m+1)a_1+(10^{m-1}+10^1)a_2+...=\sum_{k=1}^{\lfloor{\frac{m+1}2}\rfloor}({10}^k+{10}^{m-k})a_k$$ $\endgroup$ – Mythomorphic Jun 13 '15 at 14:23

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