See Exercise 4.22. Using indicators, find the variance of the number of black two-by-two subboards. Note that for the natural choice of indicators, the random variables are not independent.

Data from Exercise 4.22.

Take an n-by-n board divided into one-by-one squares and color each square white or black uniformly at random. That is, for each square flip a coin and color it white, if heads, and black, if tails. Let X be the number of two-by two square ?subboards? of the chessboard that are all black (See Fig. 4.4).

FIGURE 4.4:

Use indicator variables to find the expected number of black two-by-two subboards.