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

Enough with traveling salesman who hate long road trips.

Mario Andretti has joined the team, and he loves to drive!

His route today comprises eight towns, placed at the corners of an octagon.

Here are their coordinates:

 salesman.gif   5.9K   27 downloads

Edit: 7 and 8 are labeled incorrectly. They should be switched.

+---+-+-+

City x y

+---+-+-+

 1   0 2

 2   0 4

 3   2 6

 4   4 6

 5   6 4

 6   6 2

 7   4 0

 8   2 0

+---+-+-+

The inter-city distances, to save some calculations, are N-S and E-W distances of 6, along with four different diagonal distances of 2.828, 4.472, 5.657 and 6.083 6.325

If Mario begins driving at city 1 (coordinates 0 2) and drives until he has visited them all, how far will he be able to drive, and what city will he visit last?

If his objective is to return to his hotel in City 1 after visiting the other cities, how long of a trip could he accomplish?

You want to divide a pizza into a number of slices, but you're tired of going through the center all the time. So you adopt the strategy of marking points on the circumference of the pie and making slices along all the chords that they pairwise create. At last week's pizza party you attempted to make 32 pizza chunks, from a very large pie, by this method. How many points did you have to mark off on the pizza's circumference to give you and your 15 guests at least 2 chunks each?

Bonus: The coveted bonanova gold star will be awarded if you give the formula for calculating pieces from points.

This gives you an idea:

 pizza.gif   6.64K   21 downloads