This question tests your ability to identify the relationship between numbers and apply that relationship to find the missing number.

Let's analyze the given analogy: 42 : 20

Notice that 42 is a two-digit number. If we subtract the digits of 42 (4 - 2 = 2) and then multiply the result by 10, we get 20.

Let's apply this pattern to 64:

1. Subtract the digits of 64: 6 - 4 = 2
2. Multiply the result by 10: 2 * 10 = 20

However, this pattern doesn't seem to be consistent with the given analogy. Let's try a different approach.

Let's consider another relationship. If we reverse the digits of 42, we get 24. Now, if we subtract 4 from 24, we get 20.

Let's apply this pattern to 64:

1. Reverse the digits of 64: 46
2. Subtract 6 from 46: 46 - 6 = 40

This method doesn't match the options either.

Let's try another approach. Notice that 4 + 2 = 6 and 4 - 2 = 2. Then, if we combine the sum and the difference, we get 62. Then, subtract 42 from 62. This is not consistent.

Let's examine the correct answer. If we reverse the digits of 42 (to get 24) and then subtract 4, we get 20. If we apply this to 64, we reverse the digits to get 46, and then subtract 6, resulting in 40. This still does not work.

Let's consider another approach. The difference between 4 and 2 is 2. The result, multiplied by 10, gives 20. Applying this to 64, the difference between 6 and 4 is 2, and 2 multiplied by 10 is 20, which is still not consistent.

After exploring several approaches, let's re-examine the question. There might be an error in the question or the provided answer. None of the explored patterns consistently matches the provided analogy and answer. Further information or clarification is needed to definitively determine the relationship. The given solution of 31 seems arbitrary without a clear and consistent mathematical relationship demonstrated.