According to the first point of difference rule, second one should be correct right?
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First of all, the ‘lowest sum rule’ mentioned in the picture in the question does not exist in IUPAC nomenclature.
As already explained in a related answer, the most important simplified criteria for the numbering in such cases are:
The corresponding actual wording in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:
When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
(c) principal characteristic groups and free valences (suffixes);
(i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
(ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
(f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;
(g) lowest locants for the substituent cited first as a prefix in the name;
The compound that is given in the question doesn’t have any principal characteristic group that could be expressed as a suffix. Therefore, Rule (c) is not relevant in this case.
Thus, a low locant is assgined first to the double bond according to Rule (e). Therefore, the name of the parent structure without any further substituents is cyclohex-1-ene. (Without further substituents, however, the locant ‘1’ would be omitted.)
Finally, low locants are assigned to the remaining substituents as a set according to Rule (f).
P-14.3.5 Lowest set of locants
The lowest set of locants is defined as the set that, when compared term by term with other locant sets, each cited in order of increasing value, has the lowest term at the first point of difference; for example, the locant set ‘2,3,5,8’ is lower than ‘3,4,6,8’ and ‘2,4,5,7’.
Thus, the correct name is 1,6-dimethylcyclohex-1-ene rather than 2,3-dimethylcyclohex-1-ene since the locant set ‘1,6’ is lower than ‘2,3’.
The highest priority is double bond for this compound. And then consider to give lowest sum for substituents on double bond. Here is only one substituent present on double bond (methyl group), hence it should get the lowest number. Therefore, 1,6-dimethylcyclohex-1-ene is the correct name (1,6-dimethyl-1-cyclohexene is also correct). In either way, note that there should not have a space between 1,6-dimethyl- and cyclohex-1-ene.