| BeschreibungBalance puzzle SMIL.svg |
Interactive SVG using SMIL of a balance puzzle with inequality by CMG Lee. The five colours represent integer weights, not necessarily distinct, from 0 to 9 inclusive. In the SVG file, click the buttons until the correct code is shown.
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Let B, G, Y, R and P be the weights of the blue, green, yellow, red and purple blocks, respectively. In the rightmost balance, B = P.
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Y + G = 2 B + P, thus Y + G = 3 B.
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As Y < G and G ≤ 9 (all weights are single-digit), Y ≤ 8.
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In the second inequality from the left, Y > 2 R, thus R ≤ 3. If R were 4 or more, 2 R ≥ 8 (we established that Y ≤ 8 and Y > 2 R).
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As the left branch has B + G = Y + 2 R, we can add each side to the each side of Y + G = 3 B: (B + G) + (3 B) = (Y + 2 R) + (Y + G). Cancelling, 4 B = 2 Y + 2 R, thus 2 B = Y + R.
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Y + R must be even so that 2 B is an integer. Thus, Y and R have the same sign.
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Possible values of (R, Y) are (3, 7), (2, 6), (2, 8), (1, 3), (1, 5), (1, 7), (0, 2), (0, 4), (0, 6) and (0, 8).
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As Y + G = 3 B, G = 3 B − Y.
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Possible values of (R, Y, B, G) are (3, 7, 5, 8), (2, 6, 4, 6), (2, 8, 5, 7), (1, 3, 2, 3), (1, 5, 3, 4), (1, 7, 4, 5), (0, 2, 1, 1), (0, 4, 2, 2), (0, 6, 3, 3) and (0, 8, 4, 4).
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The only one satisfying Y < G is (3, 7, 5, 8). We can confirm Y + G = 3 B.
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Thus, B = 5, G = 8, Y = 7, R = 3 and P = 5. The code is 58735.
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