Driving man's delight - 2 a harder puzzle

[bonanova]

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
+----+---+---+

[superprismatic]

For part a: 2-5-1-6-3-7-4-8 with a distance of 51.253

For part b: 3-6-1-5-2-7-4-8-3 with a distance of 56.304

[BobbyGo]

Non Return: 1, 5, 2, 7, 3, 8, 4, 6 for 49.698

Return: 1, 5, 2, 7, 3, 8, 4, 6, 1 for 56.023

[bonanova]

Non Return: 1, 5, 2, 7, 3, 8, 4, 6 for 49.698

Return: 1, 5, 2, 7, 3, 8, 4, 6, 1 for 56.023

You're about 1.5 short on the open route but within .3 for the closed route.

Clue: try to avoid the 4-6 combination.

[tylerbrooksmusic]

1,5,2,7,4,8,3,6,1

closed

[tylerbrooksmusic]

Here's another question to tag onto the end . . . How many correct answers are there for A and B?

Edited May 16, 2013 by tylerbrooksmusic

[bonanova]

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.