Question

In a checkerboard, what is the largest number of squares that can be coloured red, so that in any arrangement of three squares at least one square is not coloured blue?

• A. 8
• B. 16
• C. 24
• D. 32
• E. 48

Answer 1

The Correct Option is D

The question is about colouring squares on a checkerboard in such a way that in any arrangement of three squares, at least one square is not coloured blue. This requires a strategic placement of colours to satisfy the condition.

A standard checkerboard has dimensions of 8x8 squares, making a total of 64 squares.

To solve this problem, we need to ensure that no configuration of three squares can all be blue. This can be achieved by colouring only half of the squares blue and the other half red.

1. Since the board is symmetrical, we will colour every alternate square, or in simple terms, we colour all the squares in one colour diagonally.
2. One simple pattern that can be followed is to colour squares like a chessboard, where no three squares that are all-coloured blue can share a diagonal or a row or column pattern.
3. If we choose to colour all the squares in a way such that each alternate square is red, it will cover half the board, i.e., 32 squares.

By this arrangement, any set of three adjacent squares will include at least one square that is not blue due to the nature of the placement.

Thus, the largest number of squares that can be coloured red, in such a manner where no three squares are entirely blue, is 32.

Therefore, the correct answer is 32.