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 3 of 6: Restrict a single light
Previous stepNext step
Lr,c(1)=Lr,c(2)=0⟺(r,c)∈{(2,2),(2,4),(3,3),(4,2),(4,4)}L^{(1)}_{r,c}=L^{(2)}_{r,c}=0\Longleftrightarrow(r,c)\in\{(2,2),(2,4),(3,3),(4,2),(4,4)\}
Detailed analysis

If exactly one cell (r,c)(r,c) is on, each invariant equals the label of that cell. Since both invariant parities are zero, the cell must have value zero in both labelings. Reading the two matrices gives exactly {(2,2),(2,4),(3,3),(4,2),(4,4)}\{(2,2),(2,4),(3,3),(4,2),(4,4)\}.