Four bells A, B, C and D begin silently. There are four strings: X changes the state of A and B; And change the one of B and C; Z changes the one of A, C and D; W changes that of D. Playing a string reverses the state of the bells that correspond to it: if they were ringing, they stop ringing; If they were silent, they start ringing. Is it possible to get only bell C to ring at the end? If you think so, indicate which ropes need to be pulled; If you think not, explain why.

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The binary bells

It seems like a mechanical problem, but the good solution is not to try combinations. It is about discovering a small quantity that never fails to obey the same rule.

Four bells A, B, C and D begin silently. There are four strings:

X changes the state of A and B;
And change the one of B and C;
Z changes the one of A, C and D;
W changes that of D. Playing a string reverses the state of the bells that correspond to it: if they were ringing, they stop ringing; If they were silent, they start ringing. Is it possible to get only bell C to ring at the end? If you think so, indicate which ropes need to be pulled; If you think not, explain why.

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Hints

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There is no need to try all 16 possible combinations.
Forget bell D for a moment and see what happens to just A, B and C.
Each string changes 0 or 2 bells there: think of parity.

Solution

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Answer: No, it is impossible. The key is to look only at bells A, B and C. Each string changes an even number of bells between those three: - X changes A and B: changes 2;

And change B and C: change 2;
Z changes A and C: changes 2;
W does not touch any of A, B and C: changes 0. Therefore, the parity of the number of bells ringing between A, B and C never changes. At first 0 of those 3 bells ring, which is an even number.

But “only C” would mean that in the end it sounds exactly 1 between A, B and C, which is odd. That can't happen. So there is no string sequence that leaves only the C bell ringing.

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