In this problem you have to determine how many carbons are attached to the indicated carbon at each square. Primary means being attached to 1 other carbon, secondary means 2, tertiary means 3 and quaternary means 4. Starting at the top with the single methyl group, since this carbon is only attached to the carbon right below it this means that it is primary. Going down 1 sigma bond, this carbon is attached to a carbon on the left, to the right, the methyl group to the top, and the carbon right below it. Since this carbon is attached to 4 carbons, this means it is quaternary. The rest of the problem continues for the same process to get the above answer.
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