The correct answer is option (D):
32
Let's break down this problem. We have a standard checkerboard, which we know is an 8x8 grid, meaning it has 64 total squares. We want to color some of these squares red, but with a restriction: within *any* three squares, at least one square must *not* be blue (meaning it is either red or uncolored).
Think about how we can approach this. We want to maximize the number of red squares, so let's try to find a pattern or structure that allows us to do this while still satisfying the condition.
The crucial observation is that we can think of the checkerboard in terms of rows and columns. Consider the smallest possible set of three squares: three squares in a row, three squares in a column or three squares that form an L shape. To guarantee at least one square is not blue, we can color all squares of the same color either red or uncolored.
Now, consider the following strategy. We can color entire rows red, avoiding any three squares being blue. How many rows could be red? Since the problem doesn't specify which arrangement of three squares, we can always ensure that our condition is met. We can colour alternate rows red. For example, row 1, 3, 5 and 7 could be colored red. Each of the four red rows contain 8 squares, therefore there are 4 rows * 8 squares per row = 32 squares colored red.
Alternatively, consider colouring alternate columns instead of rows. Rows 1, 3, 5, and 7 of each column. Again, we would have 4 * 8 = 32.
No matter how we pick three squares, at least one of them will always have a red square. Thus we cannot color any more squares red because we could then violate this condition. Let us consider if we colored 33 squares red. In the case where we can find three squares that are not red, we have violated our restriction.
Thus, the largest number of squares that can be colored red is 32.