I represent the circle in a linear manner, with (f1 f2 f3 ... fn) and an x for each eliminated friend. Each trial is separated in the parentheses by a comma.
Given 2 friends, f1 will be treated to lunch. (1 2)
Given 3 friends, f3 will be treated to lunch. (1 2 3, 4 x 5)
Given 4 friends, f1 will be treated to lunch. (1 2 3 4, 5 x 6 x)
Given 5 friends, f3 will be treated to lunch. (1 2 3 4 5, 6 x 7 x 8)
Given 6 friends, f5 will be treated to lunch. (1 2 3 4 5 6, 7 x 8 x 9 x, 10 x x x 11 x)
Given 7 friends, f7 will be treated to lunch. (1 2 3 4 5 6 7, 8 x 9 x 10 x 11, x x 12 x x x 13)
Given 8 friends, f1 will be treated to lunch. (1 2 3 4 5 6 7 8, 9 x 10 x 11 x 12 x, 13 x x x 14 x x x)
Given 9 friends, f3 will be treated to lunch. (1 2 3 4 5 6 7 8 9, 10 x 11 x 12 x 13 x 14, x x 15 x x x 16 x x)
Given 10 friends, f5 will be treated. (1 2 3 4 5 6 7 8 9 10, 11 x 12 x 13 x 14 x 15 x, 16 x x x 17 x x x 18 x)
Given 11 friends, f3 will be treated. (1 2 3 4 5 6 7 8 9 10 11, 12 x 13 x 14 x 15 x 16 x 17, x x 18 x x x 19 x x x 20)
Given 12 friends, f9 will be treated. (1 2 3 4 5 6 7 8 9 10 11 12, 13 x 14 x 15 x 16 x 17 x 18 x, 19 x x x 20 x x x 21 x x x, 22 x x x x x x x 23 x x x)
I made a graph of this data, and it seems that there isn't a pattern. Maybe I'm calculating the friend treated wrong, especially for when there are 9 friends. I guess if I didn't calculate wrong, to know where to sit you would have to memorize this data. If I was wrong and when there are 9 friends f7 is treated, you would have to memorize the pattern shown in the graph (with the point for the number of friends as 9 fixed).
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