I don't know what you mean by an interaction between 3 variables.
When you have 4 variables, you can think of them in a 2x2 table:
a b
c d
Then if you want to test if there is a difference between the change in values across the row (or column) direction, we'd calculate
(a-b) - (c-d) = a - b - c + d
But if we have 3 variables, I'm not sure how you would lay them out to test an "interaction".