bonanova

Spoiler for Slight change in the process does it

Have them stand in a straight line.Instead of requesting a show of hands, ask them to take a step forward. Liars will step forward at one or the other request. Truth-tellers from out of town will take two steps.Truth-telling locals wont' move.The one-steppers comprise all of the liars.

No need to spoiler.

If it has made a complete revolution, (and it has) it's facing the same "way" after as before.

Maybe I'm missing something.

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

on the graphic, 7 and 8 are inverted, also it seems the 6.083 distance whould have been 6.325

 

Spoiler for my take on it

not returning: 1 5 2 6 3 8 4 7 for 42.63

returning: 1 5 2 7 3 8 4 6 1 for 47.10

Xavier is correct on both counts. I've modified the OP.  Square root of 40 is 6.325.

Edit: both of these paths are just shy of optimal.

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:

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?

Spoiler for gold star - second attempt

I was to hasty with my previous answer.

Here's another try:

n(n-1)(n-2)(n-3)/24 + n(n-1)/2 + 1

That rings the bell. Well earned.

Another form is _nC₄ + _n-1C₂ + n

Sergeant McGuffy remarked to his lieutenant one day after it was all over. "The way I reconstruct the crime, the innocent people were so excited when they first talked to us, they each got one fact wrong. But, the killer wanted to confuse our trail - and so coolly lied from beginning to end." Let's recap what we know ...

Egmont VanDorn has been found dead in his apartment. From the beginning, it is pretty clear to the police that this was no accident. Before they can be separated, the four young people who found the body - Arnold, Betsy, Charles and Daisy - eagerly begin to tell their story.
 

1. "If, as the police say, Egmont was injured between four and five," said Daisy, "that must let us all out. We were all together having dinner for at least two hours before we came over here."
    
2. "But we all arrived at seven," Charles pointed out.
    
3. "Arnold said that it was six o'clock at the time, just before we opened the door. Didn't you Arnold?" said Daisy.
    
4. "I said it was just seven on the nose," said Arnold. "Sorry, honey."
    
5. "I have no idea what he said or what time we got here," declared Betsy. "The thing I remember is the gas in the hall. We rang the doorbell, and no one came and no one came, and then I smelled the gas. My heart turned over, I thought to myself, he's dead. I just know he's dead."
    
6. "Don't dramatize yourself," said Daisy coolly. "There is no way that you could have smelled the gas before we opened the door. The place was locked up, and sealed too, tighter than a drum. We'd still be in the hall if Eggy hadn't given me a key last week."
    
7. "There was gas in the hall, all right," said Charles. "I smelled it before we opened the door. You seemed to take forever getting your key out. When you finally got the door opened, the gas just streamed out."
    
8. "That was a pretty dangerous thing you did, Charles," said Arnold, "turning on the lights the way you did. Didn't it occur to you that a spark at the light switch could have blown us all up."
    
9. "The lights were already on, Arnold," Charles replied.
    
10. "For my part, I'm sorry about pulling him out of the oven - tampering with the evidence and all that," said Arnold. "Murder never crossed my mind. Locked room, you know. All I could think of was that maybe he was alive and we could still save him."
     
11. "I don't believe this," said Daisy. "You didn't pull him out of the oven. I did. You ran and opened the window. Very good move, too, I thought at the time."
     
12. "I opened the window!" cried Betsy. "I was dizzy from the fumes, and I knew I needed to do something fast."
     
13. "The only things you opened were the door to the liquor cabinet and a bottle of Scotch, Betsy." Charles laughed at her. "And I thought they were good moves."
     
14. "That was Arnold who got out the Scotch," said Betsy. "Don't you remember our sitting there after you called the police, and Arnold passing out the glasses?"
     
15. "It couldn't have been me. Must have been you," said Arnold. "I've never been here before, I didn't even know where he kept the stuff. Charles, what did you do?"
     
16. "Do you know what? I don't think I did anything. I remember quite clearly, walking over here with you after that long dinner we had together, and then seeing poor Eggy's feet through the door. But after that, I don't think I did a thing, except stand there gasping."
     

OK it's time to make an arrest.

Which liar done it?

ok, why is a circular detector bigger than the square detector ?

Circles aren't bigger or smaller than squares until constraints are added.

1. If the constraint is a given perimeter, circles are bigger (area wise)
2. If the constraint is maximum and minimum values of x and y, then squares are bigger.

The present constraints are of the second type.