Number the unit squares with their "time when a checker has been put" there (For convenience, I will call it as "clicking"). For example, write our initial square 1, the second clicked square 2, and so on. Draw a directed edge from each unit square to its neighbor if its neighbor has a bigger number than it (For example, ,if 3 and 5 are neighbors, draw a directed edge from 3 to 5). Observation: The sum of squares' outer degrees is even. Proof: We have a total of $2n(n-1)$ edges and as each edge is counted once, the proof is completed. Assume the contrary, then note that the outer degree of 1 is odd, and for any other number $i$, the outer degree of $i$ should be even (Note that only the outer degrees of $i$ affect the colour of $i$). Thus the sum of all outer degrees is odd, which is a contradiction.