  A cryptarithm is a mathematical puzzle where given an arithmetic equation
written with letters rather than numerals, a player must discover an assignment
of numerals on letters that makes the equation hold true. In this paper, we
propose a method to construct a DFA that accepts cryptarithms that admit
(unique) solutions for each base. We implemented the method and constructed a
DFA for bases $k \le 7$. Those DFAs can be used as complete catalogues of
cryptarithms,whose applications include enumeration of and counting the exact
numbers $G_k(n)$ of cryptarithm instances with $n$ digits that admit base-$k$
solutions. Moreover, explicit formulas for $G_2(n)$ and $G_3(n)$ are given.
