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I owe it to the Den to post at least one of these without error. Possibly this one does that. Give the longest route (sequence of city numbers) that visits all the cities (a) not returning to starting city (sum of 7 distances - starting point matters) (b) returning to starting city (sum of 8 distances - starting point does not matter) This puzzle has more choices than the first one. Cities lie clockwise on the perimeter of a 6x6 square: 6--O------O-------+--------O |3 4 5| | | | | 4--+ + | | | | | | 2--O2 6O | | | | |1 8 7| 0--O-------+-------O-------O | | | | 0 2 4 6 +----+---+---+ |City| x | y | Distances: +----+---+---+ 8.485 1-5 3-7 | 1 | 0 | 0 | 7.211 2-5 3-6 3-8 4-7 | 2 | 0 | 2 | 6.325 1-4 1-6 2-7 4-8 5-8 | 3 | 0 | 6 | 6.000 1-3 1-7 2-6 3-5 5-7 | 4 | 2 | 6 | 5.656 4-6 | 5 | 6 | 6 | 4.472 2-4 2-8 | 6 | 6 | 2 | 4.000 1-8 2-3 4-5 5-6 | 7 | 6 | 0 | 2.828 6-8 | 8 | 4 | 0 | 2.000 1-2 3-4 6-7 7-8 +----+---+---+

I heard the announcer shout excitedly as the ball cracked off the bat. "Going, going, gone!" he cried. My face and and the 12 stitches embedded in it beg to differ about that last part.