Let's consider two matrixes nXn - a and b, and two array c,d :[1..n]
The first has a[i,j]=1 if person i switches valve j, and 0 if doesn't.
Initially, b[i,j]=1 if and only if i=j. Further this matrix will
contain the information about how did we get the corresponding row of
matrix a (I'll be more detailed about this below)
Initially, c[i]=0 for all i. Further, c[i]=1 means that for current
moment valve number i is turned on.
Initially, d[i]=0 for all i. This array will have d[i]=1 if we used i-
th person for current moment.

Then, we shall use Gauss method with XOR operation in order to make
string C consist only '1', using  XOR for lines of a and for line c,
doing the same actions with corresponding lines of b and line d.


I think that's enough for you to solve the problem.

Good luck!

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