Can deterministic Turing machine beats/wins (if possible) the "Bombs and Levers" game in polynomial time?
Your problem is NP-hard. It is an easy proof In fact, it is NP-complete because we can reduce it to SAT in polynomial time

We reduce 3-SAT to your problem.

We have a lever for each variable and a bomb for each clause, the red cables are connected to clauses/bombs that have positive literals belonging to that variable/lever and the green cables are connected to clauses/bombs that have negative literals belonging to that variable/lever.

Each bomb has the number 3 written on it, meaning that the bomb will explode if it is decremented three times. Whenever we set a lever that is connected to the bomb in the direction that doesn´t deffuse the bomb we decrement the counter in the bomb. This is the same as setting a variable true if the literal in a clause is false or viceversa. If we do this three times that means that the clause has not being satisfied in the original 3-SAT problem and the bomb explodes.

If the instance is satisfiable, you can always choose the right positions for the levers so that no bomb explodes. If there is a variable assignment that satisfies the clauses that means that the bombs can be defused. A defused bomb cannot explode anymore so that we don´t have to worry about a defused bomb anymore.

So, can your problem be solved in polynomial time? We don´t know. People will have to give a solution to the P vs NP problem first, come back in a hundred years and maybe we will have an answer