Problem 5

A regular

5\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 4 of 6: Realize the first position

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T_{2,2}=\{(1,2),(1,4),(2,1),(2,2),(2,4),(2,5),(3,4),(4,1),(4,2),(4,3),(5,2)\}

Detailed analysis

Toggle exactly the cells in $T_{2,2}$ (coordinates are row, column). Directly adding the local crosses modulo 2 leaves only $(2,2)$ on. Rotating the whole toggle set gives the analogous construction for $(2,4)$ , $(4,2)$ , and $(4,4)$ .