Alright, let's get started. The task is to look for pairs of numbers that add up to 6 amidst the pairs listed.
Let's evaluate each pair one after the other.
1. -7 and 1: Summing these two, -7 + 1 is -6, not 6. So, this pair doesn't fit our criteria.
2. 12 and -6: If we add these two, 12 + -6 equals 6, which fulfills our requirement. Thus, this pair is a valid pair.
3. -9 and 3: Adding these, -9 + 3 is -6, not 6. So, this pair doesn't satisfy our criteria.
4. 2 and -8: Summing the pair 2 + -8 equals -6, not 6. Consequently, it doesn't fit our criteria.
5. -4 and 10: And finally, if we sum these two, -4 + 10 equals 6. Therefore, this pair meets our criteria.
So from our assessment, only the pairs (12, -6) and (-4, 10) add up to give the number 6.