MathLabs

Problem 5

A regular 5×55\times5 array of lights is defective: toggling the switch for one light causes each adjacent light in the same row and in the same column, as well as the light itself, to change state. Initially all lights are off. After some toggles, exactly one light is on. Find all possible positions of this light.
Step 5 of 6: Realize the central position
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T3,3={(1,2),(1,3),(1,5),(2,1),(2,5),(3,1),(3,3),(3,4),(4,3),(5,1),(5,2)}T_{3,3}=\{(1,2),(1,3),(1,5),(2,1),(2,5),(3,1),(3,3),(3,4),(4,3),(5,1),(5,2)\}
Detailed analysis

Toggle the cells in T3,3T_{3,3}. Checking the five-cell local-cross rule modulo 2 leaves exactly the central cell (3,3)(3,3) on. Thus the fifth common-zero position is attainable as well.