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A1 must be diagonally adjacent to a red square. The only square that is diagonally adjacent to A1 is B2, so B2 is red.
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A5 must be diagonally adjacent to a square of the same color. The only square that it is diagonally adjacent to is blue, so A5 is blue.
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Since every corner is adjacent to all 3 colors, each square that is adjacent to a corner must be different from the other two squares adjacent to the same color. Thus A2 and B1 contain green and blue, but we do not know which is which yet. But since we know that they are different, we know A2 has to be the same as B3 to stay diagonally adjacent to a square of the same color. Thus B3 must be green or blue.
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Using similar logic around the top right corner, we know A4 and B5 are red and green. Also B3 needs to be the same as A4 because we know B5 is different from A4. Thus B3 must also be red or green. Since we previously said it was green or blue, we can deduce that B3 must be green. It follows easily that A2 and A4 are also green.
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In order for A1 to be adjacent to a blue square, B1 must be blue. C2 is also blue because B1 needs to be diagonally adjacent to square of the same color. Similarly, B5 and C4 are red.
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Similar to steps 3 and 4, D1 is different from from E2 and is thus congruent to C2. D5 is different from from E4 and is thus congruent to C4.
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We know E2 is red or green. We know E2 is diagonally adjacent to a square of the same color. That square must be D3. Thus D3 is red or green. We know E4 is blue or green. We know E4 is diagonally adjacent to a square of the same color. That square must be D3. Thus D3 is blue or green. Since it we previous asserted that D3 was red or green, D3 must be green. It follows that E2 and E4 are also green.
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Since the E1 corner needs to be adjacent to a red, D2 is red. Since E1 needs to be diagonally adjacent to a square of the same color, E1 is also red. Similarly D4 and E5 are blue.
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C1 needs to be diagonally adjacent to a square of the same color, so it must be red. Similarly, C5 is blue.
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The 3 squares in the 3rd column are not red, and are not diagonally adjacent to any green, so all 3 must be blue.