The first person must take either stool 9 or 17
(because of symmetry, it doesn't matter which). Assume they pick seat
9. The next person will pick seat 25, since it is the furthest from
seat 9. The next two people will take Seats one and 17. The next three
will occupy 5, 13, and 21. The next six will occupy 3, 7, 11, 15, 19,
and 23. This seats the maximum of 13 people, and no one is sitting next
to another person. If a seat other than 9 or 17 is chosen first, the
total bar patrons will be less than 13.
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