| DescriptionBalance 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.
| Solution |
|---|
|
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.
| Next hint… |
|---|
|
Y + G = 2 B + P, thus Y + G = 3 B.
| Next hint… |
|---|
|
As Y < G and G ≤ 9 (all weights are single-digit), Y ≤ 8.
| Next hint… |
|---|
|
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).
| Next hint… |
|---|
|
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.
| Next hint… |
|---|
|
Y + R must be even so that 2 B is an integer. Thus, Y and R have the same sign.
| Next hint… |
|---|
|
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).
| Next hint… |
|---|
|
As Y + G = 3 B, G = 3 B − Y.
| Next hint… |
|---|
|
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).
| Next hint… |
|---|
|
The only one satisfying Y < G is (3, 7, 5, 8). We can confirm Y + G = 3 B.
| Next hint… |
|---|
|
Thus, B = 5, G = 8, Y = 7, R = 3 and P = 5. The code is 58735.
|
|
|
|
|
|
|
|
|
|
|
|
