The Seam is actually different from District 12. The Seam is an area within District 12, as well as the poorest. It's mostly a mining town, with more miners than merchants. It's possible that it's called the Seam because the district is located in Appalachia, and the Appalachian Mountains look similar to a seam on the continent.
Another possibility is that the Seam is located right on the border between District 12 and the wilderness, forming a 'seam' between the two. The word "seam" literally means "where two pieces of fabric are sewn together." So, this area of District 12 could be considered where the oppressed country and the slightly rebellious and very free woods come together. A seam is often more blended than a rigid border, which is represented by the broken electric fence, and how easy it is for people of District 12 to go between the wilderness and the safety of the district.
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At the beginning of 'The Hunger Games', Katniss lives in an area of District 12 called 'The Seam', where the less-wealthy inhabitants live. Later, she gains a large home in Victor's Village.
They live in the country of Panem, in District 12.
Katniss Everdeen is from district 12, which is the coal mining district.
Katniss was born in the seam and lived there until she won the hunger games
Most people in the Seam work in the coal mines of District 12, but some are beggars. Also, many work as black market vendors in the Hob, the flea market where Katniss sells her wild game.