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 6 of 6: Conclude the list
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PossiblePositions⁡={(2,2),(2,4),(3,3),(4,2),(4,4)}\operatorname{PossiblePositions}=\{(2,2),(2,4),(3,3),(4,2),(4,4)\}
Detailed analysis

The invariants rule out every other cell, and the explicit toggle sets realize all five listed cells. Therefore the possible positions are exactly {(2,2),(2,4),(3,3),(4,2),(4,4)}\{(2,2),(2,4),(3,3),(4,2),(4,4)\}.