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ISSN Print: 1064-2315
ISSN Online: 2163-9337

# Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v51.i7.30
pages 34-46

## Methods of Solving the Problems of Mathematical Safe on Elementary Graphs

Artem L. Gurin
National Technical University of Ukraine "Igor Sikorsky Kiev Polytechnic Institute", Kiev
Andrey G. Donets
National Aviation University, Kiev
Sergiy Zagorodnyuk
Taras Shevchenko National University of Kyiv

### ABSTRACT

The problem of mathematical safe consisting of a certain system of interrelated locks with given initial states is under consideration. Such system can be represented in the form of oriented or non-oriented graph, which vertexes are locks. In this paper, we consider graphs of sufficiently simple structures, such as path, contour, chain, cycle, fantail, stairs with a prescribed quantity of steps, and complicated stairs. In the general case, solution of this problem is reduced to solving a system of linear equations in the class of subtracts in absolute value, which is equal to the number of states of every safe lock. In fact, this number equals to the number of key turns in each lock for reaching finally transition of the safe in the state when all locks are open. To solve this problem, two original methods are suggested, namely, the method of separation of variables and the method of summary representations. The essence of first method consists in the following. For some elementary graphs there is a potential of singling out some equations for their immediate solving relative to one arbitrary variable. Further, substituting successively the obtained solutions into the corresponding equations, we obtain the solution of the system. This method was used for solving the problem for the graph of a cycle type. The essence of the second method consists in the introduction of a special parameter called the sum of unknowns. Some graphs make possible to present the system variables by this parameter. Summing these variables we obtain the equation relative to this parameter. Having solved this equation we obtain the value of this parameter as well as the values of all variables. This method was used for solving the problem for the graphs of window and stairs types. Every problem for the prescribed types of safe is shown by examples and is supplemented by verification of the solution.

### REFERENCES

1. Donets G.A., Solution of safe problem on (0, l)-matrices, Cybernetics and Systems Analysis, 2002, No. 1, 98-105. .

2. Agai Ah Garnish Yakub, Donets G.A., Problem of mathematical safe on matrixes, Teoriya optimalnykh resheniy, 2013, 124-130. .

3. Kryvyi S.L., Algorithms for solution of systems of linear Diophantine equations in residue fields, Cybernetics and Systems Analysis, 2007, 43, No. 2, 171-178. .

4. Donets G.A., Gurin A.L., Problem about mathematical safe oflocks with two states, Mezhdunarodnyi nauchno-tekhnicheskiyzhurnal "Problemy upravleniya i informatiki", 2018, No. 5, 33-41. .

5. Kryvyi S.L., Solution algorithms for systems of linear equations over residue rings, Cybernetics and Systems Analysis, 2016, 52, No. 5, 149-160. .

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