Gauche correction in Bensen group increment
Here is how I do it.


Sight down each C-C bond along the backbone of the molecule and draw a Newman projection
Look at all 3 possible conformations in the Newman projection and pick the conformation with the fewest gauche interactions
Count the number of gauche interactions
Repeat with the next C-C bond along the backbone


Here is a drawing of the 3 Newman projections in 3,4-dimethylhexane that contain gauche interactions (there are no gauche interactions along the $\ce{C1-C2}$ or $\ce{C5-C6}$ bonds.  You can see that there are 4 gauche interactions in the molecule.