let's assume that a1, a2, a3,...an are will mean the relation of first mechanic to valves,
b1, b2, b3, b4...bn are relation of second mechanic to valves, i.e if a1 = 1 then first mechanic must switch the first valve, else no.
let's assume that x1  means whether mechanic to send to switch his valves, x2 for 2-nd mech, so on...
then it's obvious that
a1 * x1 + a2 * x2 + a3 * x3..... = 1 mod 2
b1 * x1 + b2 * x2+ b3 * x3 .... = 1 mod 2
,......
this Linear system equation can be solved by Gaussian algorithm
we need to find only x1, x2, x3,,,,
where if xi (1<= i <= n) is 1 then we will send him, otherwise no.
if the determinant is 0, then No solution...