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As title reads, why shouldn't 1-methyl-hexane work but 2-methyl-hexane and 3-methyl-hexane does?

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You need to always number your longest chain first. If you draw 1-methylhexane, you'll see the longest chain isn't actually 6 carbons anymore, but 7. Then all the atoms are part of the longest chain, and the name simply becomes heptane.

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