TY - JOUR

T1 - Enumerating cryptarithms using deterministic finite automata

AU - Nozaki, Yuki

AU - Hendrian, Diptarama

AU - Yoshinaka, Ryo

AU - Horiyama, Takashi

AU - Shinohara, Ayumi

N1 - Publisher Copyright:
Copyright © 2018, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2018/7/26

Y1 - 2018/7/26

N2 - 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 ≤ 7. Those DFAs can be used as complete catalogues of cryptarithms, whose applications include enumeration of and counting the exact numbers Gk(n) of cryptarithm instances with n digits that admit base-k solutions. Moreover, explicit formulas for G2(n) and G3(n) are given.

AB - 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 ≤ 7. Those DFAs can be used as complete catalogues of cryptarithms, whose applications include enumeration of and counting the exact numbers Gk(n) of cryptarithm instances with n digits that admit base-k solutions. Moreover, explicit formulas for G2(n) and G3(n) are given.

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M3 - Article

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