The analysis says that whenever you are left with a connected blank space (not 
necessarily rectangular) whose size is divisible by X - then you can fill it up 
with X-ominoes:

" If a connected blank area of size M is a multiple of X, it can be guaranteed 
that there is a way to place M/X X-ominoes to fill in the blank area. "

It also repeats later.

It seems that what is actually used in the proof is the claim in the other 
direction: if you can force a blank space to be of size not divisible by X then 
you win (no way to fill the blank space).

The claim as stated doesn't seem quite right actually. Consider the case of a 
connected blank space of size 4 whose shape is of one blank space with three 
other blank spaces around it (adjacent to it). Then obviously there is no way 
to fill this space with 2 2-ominoes. 
The claim in the analysis is not quite right. Maybe it can be right with more 
restrictions and in a specific content, but not the way it is currently claimed.

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