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# Counting Problem

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How many ways are there to color the 4x4 grid so that each unit square is red, blue, green or yellow, and so that each unit square is the same color as exactly two of the unit squares that share a side with it?

May 14, 2024

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To solve this problem, let's consider each square individually. Each square must have two adjacent squares of the same color. Let's count the possibilities for each square:

1. **Corner squares (4 squares):**

These squares have only two adjacent squares. Thus, they must be the same color as those adjacent squares. So, each corner square has 2 choices of color.

2. **Edge squares (8 squares):**

These squares have three adjacent squares. Again, they must be the same color as two of these adjacent squares. So, each edge square also has 2 choices of color.

3. **Interior squares (4 squares):**

These squares have four adjacent squares. They must match two of these adjacent squares. Each interior square has 2 choices of color.

So, the total number of ways to color the grid is the product of the choices for each square:

$\text{Total} = 2^4 \times 2^8 \times 2^4 = 2^{16}$

Therefore, there are $$2^{16} = 65,536$$ ways to color the 4x4 grid according to the given conditions.

May 14, 2024