Answer to Puzzle 1: We will ignore the curvature of the earth, quibbles about quantum uncertainty, or other arguments. If you look at a common map of the United States, then for every state, you will be able to identify a straight line segment that is part of the boundary, except for Hawaii. Therefore, the answer is 2%.
Answer to Puzzle 2: You may be able to visualize a map of the west, and realize that Colorado and Wyoming both have rectangular shapes formed by 4 lines. The state of New Mexico, has a more complicated shape, but seems also to be formed only from straight lines, but there's a bit of border with Texas that is crooked. That leaves Utah as the third state that is formed from (6) straight lines. Therefore, the answer is 6%.
Answer to Puzzle 3: The border between the US and Canada extends from Washington State to Minnesota, following the 49th parallel of latitude. However, when this border reaches the Lake of the Woods, it jogs northward and includes a portion of land that is reachable from the US only by crossing the lake. The border then extends to the east a certain distance before dropping south below the 49th parallel and continuing eastwards. Despite the fact that Maine seems to jut back up above the 49th parallel in some maps, this is not the case. So the territory of Minnesota known as the Angle or the Northwest Angle is more northerly than any other point in the US outside of Alaska.
Answer to Puzzle 4: You can point south to Canada from Alaska (no surprise). You can also do it from a tiny bit of Washington State, not connected to the rest of the state, which "overhangs" Vancouver Island. You can also do it in a large portion of Michigan, north of Detroit, which "overhangs" Ontario. On a smaller scale, there are bits of the Minnesota border, such as near Voyageurs National Park, which overhang bits of Canada. Drummond Island in Michigan overlaps a bit of Cockburn Island. Near Niagara Falls, the border between New York and Ontario includes a significant overhang. Finally, the border of Maine includes a number of indentations that result in overhangs.
Answer to Puzzle 5: Although they are not actually contenders, we should decide how to deal with the so called "Four Corners States", namely, Colorado, New Mexico, Arizona, and Utah, whose boundaries meet at one point. Technically, for instance, we would say that Arizona and Utah are not neighbors, because their borders only touch at one point, not along a significant line segment.
Now we note the sad facts that both Hawaii and Alaska have zero neighbors.
Now note that the long skinny state of Tennesseee hits the jackpot with 8 neighbors, Virginia, Kentucky, Missouri, Arkansas, Mississippi, Alabama, Georgia, and North Carolina.
This result is tied by Missouri, whose eight neighbors are Kentucky, Illinois, Iowa, Nebraska, Kansas, Oklahoma, Arkansas and Tennessee.
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