The analogy presented is based on a mathematical relationship between the two numbers in each pair. Let's analyze the given pair (3:31):

Notice that \(3^3 + 2 = 27 + 2 = 29\). This is not directly equal to 31. Let's explore other possibilities. Let's consider the pattern \(x : (x^3 + 2x)\).

If we apply this to the given pair (3:31):

• \(3^3 + (2 \times 3) = 27 + 6 = 33\). This is not exactly 31.

Let's try another approach. Let's consider the pattern \(x : (x^3 + x + 1)\).

Applying this to (3:31): \(3^3 + 3 + 1 = 27 + 3 + 1 = 31\). This matches!

Now, let's test this pattern with the other options:

• Option 1 (6:220): \(6^3 + 6 + 1 = 216 + 6 + 1 = 223 \neq 220\). This doesn't quite fit.
• Option 2 (4:65): \(4^3 + 4 + 1 = 64 + 4 + 1 = 69 \neq 65\).
• Option 3 (5:128): \(5^3 + 5 + 1 = 125 + 5 + 1 = 131 \neq 128\).
• Option 4 (8:315): \(8^3 + 8 + 1 = 512 + 8 + 1 = 521 \neq 315\).

Let's reconsider the pattern. Let's try another pattern: \( x : (x^3 + 2x - 1)\). Applying this to (3,31): \(3^3 + 2(3) - 1 = 27 + 6 -1 = 32\).

Let's try another pattern: \( x : (x^3 + 4x - 2)\). Applying this to (3,31): \(3^3 + 4(3) - 2 = 27 + 12 - 2 = 37\).

Let's try \( x : (x^3 + 2x + 1)\). Applying this to (3,31): \(3^3 + 2(3) + 1 = 27 + 6 + 1 = 34\).

There might be a slight discrepancy in the question or the options provided. However, based on the closest fit, we need to re-evaluate the pattern. A more suitable pattern might exist for the given options. Let's reconsider option 1 (6:220). If we use the formula \(n : (n^3 + n^2 - n)\), then for n = 3, we get 33, which is close to 31. For n = 6, we get \(6^3 + 6^2 - 6 = 216 + 36 - 6 = 246\), which is not 220. Further investigation is needed to find the exact pattern.

Based on the provided options and the reasoning above, further clarification is required on the intended pattern or there might be an error in the question or options.