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 2 of 6: Rotate the labels

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L^{(2)}=\operatorname{rot}_{90^\circ}(L^{(1)})

Detailed analysis

Rotate $L^{(1)}$ by $90^\circ$ to obtain $L^{(2)}$ . The same local-cross check shows that the second weighted value also has invariant parity under every toggle. Both values start with parity zero because initially all lights are off.