The Antisocial Club Challenge

The Antisocial Club meets every week at Jim's Bar. Since they are so antisocial, however, everyone always sits as far as possible from the other members, and no one ever sits right next to another member. Because of this, the 25-stool bar is almost always less than half full and unfortunately for Jim the members that don't sit at the bar don't order any drinks. 

Jim, however, is pretty smart and makes up a new rule: The first person to sit at the bar has to sit at one of two particular stools. If this happens, then the maximum number of members will sit at the bar. 

Which stools must be chosen? Assume the stools are numbered 1 to 25 and are arranged in a straight line.

Wise Selection of first 2 Seats made available! 

Logical Response to The Antisocial Club Challenge

What was the challenge?

The bar owner Jim put only 2 options for first person to seat and those options are Stool No. 9 and Stool No.17. We'll call seat for stool for convenience.

Choice of numbers 9 and 17 is because of symmetry explained below.

Suppose the first person choose seat no.9.

Then the second person has to choose the furthest seat i.e. seat no.25.

The third person will choose seat no.1 which is furthest from seat no.9 and 25.

The fourth person won't have option other than seat no.25 which is the furthest from rest of 3.

For the fifth person, seats numbered 5, 13 and 21 are available which are equidistant from 2 persons. We assume he chooses farthest from the seat occupied by person who entered just ahead of him. That is he chooses seat no.5.

Since, seat no.21 is furthest from seat no.5 and only seat equidistant from 17 & 25, the sixth person will choose seat no.21.

With that, 7th person won't have any option other than seat no.13 to maintain distance from others.

With the same logic, next 6 persons occupies seat numbered 3, 23, 7, 19, 11, 15.

So there will be 13 people maximum in the bar with no one seating next to each other.

The same seat numbers will be occupied even if the first person choose seat no.17. The second person will choose seat no.1, third will choose seat no.25, fourth will choose seat no.9 and so on.

But if the first person doesn't choose 9 or 17 then there can be less than or equal to 12 people maximum in the bar following their antisocial trait.