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The city is famous for its Nickel Mining and one major local attraction is the Giant Nickel 5cents, which is located on the western side of the city (near Route 55 and Lorne Street).
West to Sault Ste. Marie
From Sudbury get to any entrance of the Trans Canada Highway. Sudbury is very spread out with few buses so walk/hitch to the closest entrance. Hitchhiking to Sault Ste. Marie in the winter: https://www.youtube.com/watch?v=aSqKOsjVETs
From Downtown where the train station is, it's easiest to walk down Elgin Street and then Lorne.
East to North Bay
There is a spot next to the onramp at the intersection of the Highway 17 with the 55. Plenty of room for cars to stop.