Question 2
There is a lamp on each cell of an infinite square grid. Initially, all lamps are *off*. A move consists of choosing either a $3 \times 3$, $4 \times 4$ or $5 \times 5$ square contained on the grid and switch all lamps inside that square from *on* to *off* or vice versa.
a. Prove that for any finite set $S$ of lamps it is possible to achieve, after a finite sequence of moves, that all lamps that are *on* are exactly the ones in $S$.
b. Prove that if a sequence of moves only involves two of the three sizes of squares available, then it is not possible to achieve that in the end all lamps that are *on* are exactly the ones inside a $2 \times 2$ square.
a. Prove that for any finite set $S$ of lamps it is possible to achieve, after a finite sequence of moves, that all lamps that are *on* are exactly the ones in $S$.
b. Prove that if a sequence of moves only involves two of the three sizes of squares available, then it is not possible to achieve that in the end all lamps that are *on* are exactly the ones inside a $2 \times 2$ square.