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Posted: Wed Feb 01, 2017 2:50 pm
by Kazi_Zareer
A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that no matter what the starting
positions are, it is always possible to open this lock. (Only one key at a time can be switched.)


Posted: Wed Feb 01, 2017 3:31 pm
by ahmedittihad
Nice problem.
Mark a horizontal key and switch every key on the column and row of the marked key including the marked key once. This results in only changing the state of the marked key. Thus we can switch every horizontal key vertical.