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There is a 2x2 matrix lock with one key in each cell. A key is nothing but a straight lever which can either be either in vertical or horizontal state. Please see the attached picture. The lock opens only when all the levers are horizontal. However, when you change the state of the lever in a cell, say cell A, the levers of all the cells in the same row and column as cell A switch their states. The question is if we can start with the levers in any state and open the clock by turning the levers.

I am able to solve this mechanically, irrespective of the state of levers at the beginning. However I want to be able to prove it using some math, but not sure how I can go about it. Can anybody please help me?

This puzzle is a subset of a bigger puzzle involving a 4x4 matrix. I think if I learn to prove how the 2x2 lock can always be solved, I can extend it to the 4x4 lock.

Thanks,

MG.

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# Lock problem -need a mathematical proof

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