In which problems we can use solving System of algebraic linear equations, such as by Gauss' method?

Is there any problems(not theoretical)?

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You can also solve it by Gauss when minimum value is required (assuming that you must take at least one A, otherwise the answer is always 0).

Problem 2: There was also one on Topcoder but I can't remember which SRM it was so here is a brief description for the problem statement:

You have N bulbs and N buttons. At the beginning all the bulbs are switched off. Each button can change the state of some of the bulbs. Given N and the bulbs each button affects, return the least number of buttons needed from these buttons such that you can reach all the possible configurations you can reach using any combination of the N buttons. (1 <= N <= 50).

Problem 3: There is also the direct type:

http://livearchive.onlinejudge.org/index.php?option=com_onlinejudge&page=show_problem&problem=2742

P.S. In the last one the sample input is incorrect, here is the correct one:

Simliar one: Square

Besides, LIM of SPOJ is a cyclic probability problem. NWERC04H, which should be solved using gaussian elimination modulo P.

They all use some sort of modulo field.

There's another Markov Chain kind of problem -> MARIOGAM, also on spoj.