The “explanations” in this problem come strictly from algebraic proofs.
Take the product of the middle two numbers and subtract the product of the first number and the last number.
Example: 5, 7, 9, 11
Middle Product: 63
Outer Product: 55
63 – 55 = 8
Example: -9, -7, -5, -3
Middle Product: 35
Outer Product: 27
35 – 27 = 8
Conjecture: This difference will always be 8.
Consider four consecutive odd numbers:
(2k+1), (2k+3), (2k+5), (2k+7), where k is an integer.
Does the same thing happen for even numbers?
Indeed it does. Consider the following four consecutive even integers:
2k, 2k+2, 2k+4, 2k+6, where k is an integer.
What happens for consecutive integers in general?
Consider the following consecutive integers:
k, k+1, k+2, k+3, where k is an integer
When the two are subtracted, 2 is always the result.