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Driving man's delight - 2 a harder puzzle


bonanova
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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
+----+---+---+

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SP is first with both solutions.

Tylerbrooksmusic was first with the closed course solution

Good job both.

The critical part of both solutions involves cities 3 4 7 and 8.

They have to be taken as 3 7 4 8 in the open case (a) and 3 8 4 7 in the closed case (b).

In (a) you need cities 2 and 8 as endpoints: because 2-8 is the shortest path that you could add to make the path closed.

In (b) you need to be at city 7 to make the next link to city 2 as long as possible (384725...) instead of (473825....)

Once you fix those sequences, I think the remainder of the two paths are determined.

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