МЕТОДИ РОЗВ’ЯЗУВАННЯ ЗАДАЧ ПРО МАТЕМАТИЧНИЙ СЕЙФ НА ЕЛЕМЕНТАРНИХ ГРАФАХ

• Datum: 2025
• Hauptverfasser: Gurin, A.L., Donets, A.G., Zagorodnyuk, S.P.
• Format: Artikel
• Sprache: English
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Schlagworte: математичний сейф; вектор станів замків сейфа; неорієнтований граф; шлях; контур; ланцюг; цикл; віяло; драбинки двох типів та ускладнені драбинки; метод виділення змінних; метод сумарних представлень
• Online Zugang: https://jais.net.ua/index.php/files/article/view/666
• Назва журналу: Problems of Control and Informatics

We consider the problem on mathematical safe, consisting of certain system of interrelated locks with given initial states. Such system can be presented in the form of oriented or non-oriented graph, which tops are locks. In this paper we deal with graphs of sufficiently simple design, such as path, contour, chain, cycle, umbrella, stairs with prescribed quantity of steps, and complicated stairs. In general case, solution of this problem reduces to solving a system of linear equations in the class of residues in modulo, which equal to the number of states of each safe lock. In fact, it equals to the number of key turns in each lock, sufficient for safe to switch info the state with all open locks. To solve this problem, two original methods are suggested, namely, the method of variables separation and the method of combined representations. The gist of first method consists in the following. For some elementary graphs some equations can be separated and solved in certain variable. Then, upon successive substation of obtained solutions into corresponding equations, we come to solution of the system. This method was applied to solve the problem for the graph of cycle type. The gist of second method consists in introduction of special parameter, called the sum of unknowns. For some graphs, to present the system variables through this parameter is a possibility. Upon summing these variables, we obtain equation in the mentioned parameter. We add these variables and come to the equation in this parameter. Upon solving this equation we obtain value of this parameter as well as the values of all variables. This method was applied to solve the problem for the graphs of window and complicated stairs types. Each problem for prescribed safe types is illustrated by examples and accompanied by solution verification.