Question

The reaction of chlorine (Cl2) with 1,4-dimethylcyclohexane (show three possible monochlorination products) *what?!*

Answer 1

I think the tertiary carbons are most reactive followed by secondary then primary.

Answer 2

I would start be defining which which 1,4-dimethylcyclohexane I was starting with, or considering a mixture of cis and trans?

The tertiary carbons are most reactive as DrBob says, so there are two possible products from attack at the tertiary carbon:
cis and trans 1-chloro-1,4-dimethyl cyclohexane.

Attack at the secondary carbon adjacent to one of the methyl groups will result in 2-chloro-1,4-dimethylcyclohexane. If you start with a mixture of the cis and trans there are eight possible products which are:

1R,2R,4R, 1-chloro-1,4-dimethyl cyclohexane.
1S,2R,4R, 1-chloro-1,4-dimethyl cyclohexane.
1R,2S,4R, 1-chloro-1,4-dimethyl cyclohexane.
1R,2R,4S, 1-chloro-1,4-dimethyl cyclohexane.
1R,2S,4S, 1-chloro-1,4-dimethyl cyclohexane.
1S,2R,4S, 1-chloro-1,4-dimethyl cyclohexane.
1S,2S,4R, 1-chloro-1,4-dimethyl cyclohexane.
1S,2S,4S, 1-chloro-1,4-dimethyl cyclohexane.

Attack at the primary carbon in one of the methyl groups will give
1-chloromethyl-4-methylcyclohexane. Again you have the possibility of cis and trans depending on the starting material.

Answer 3

There is of course a deliberate mistake. I meant to list the eight as

1R,2R,4R, 2-chloro-1,4-dimethyl cyclohexane

etc.