To solve the problem of finding the number of valid 3x3 grid colorings (red/blue) where every 2x2 subgrid has exactly two red and two blue cells, we analyze the constraints and split into cases:

Case 1: Constant Rows

Row 1 is all red (RRR) or all blue (BBB) → 2 options.

• Row 2 must be flipped (BBB/RRR) to avoid invalid 2x2 subgrids.
• Row 3 must be flipped again (back to row1) → only 1 choice.
  Total: 2 × 1 × 1 = 2 grids.

Case 2: Alternating Rows

Row1 is alternating (RBR or BRB) → 2 options.

• Row2 can be same as row1 or flipped → 2 choices.
• Row3 can be same as row2 or flipped → 2 choices.
  Total: 2 × 2 × 2 = 8 grids.

Case3: Non-constant, Non-alternating Rows

Row1 is non-constant and non-alternating (RRB, RBB, BRR, BBR) → 4 options.

• Row2 must be flipped (only 1 choice) to avoid invalid subgrids.
• Row3 must be flipped again (back to row1) → only 1 choice.
  Total:4 ×1 ×1=4 grids.

Total Valid Grids

Summing all cases: 2 +8 +4 = 14.

Answer: 14.

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