A 13×13 grid of lights can be controlled according to a series of switches. For any 9×9 or 2×2 square of lights there is a switch that reverse each of those 9^2or 2^2 lights. Initially, all 13^2lights are off. Determine whether or not it is possible to achieve every lighting configuration using some combination of switches.
I am trying to divide the 169 squares into 2 sets of 9^2 unit squares and then proceed with the rest 7 squares but not being able to do so please help
Throwing a switch twice makes no sense because it flips the bulbs twice and the switch flips commute.
There are $169$ switches, so $2^{169}$ combinations of switches you might throw.
There are $2^{169}$ patterns, so if we can reach any pattern there must not be any pattern we can reach in more than one way.
If we switch the $9 \times 9$ block in each corner the central $5 \times 5$ block gets switched four times and is off. The $4 \times 5$ blocks in the center of each side get switched twice and are off. The $4 \times 4$ block in each corner is switched once and is on. You can flip the corners using $2 \times 2$ flips and return to all lights off. We have found a set of flips that does nothing, so we cannot reach all the patterns.