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Since each person has 2 neighbors at a time, and there're 6 people other than him, there are 3 arrangements at most.(otherwise colliding arrangements must exist)
now let's find out a possible tuple of arrangements, by selecting along the circle, skipping 0,1 and 2 people each step.
note that an equivalent condition is that each one of the nCr(7,2)=21 possible pair of neighbors exist at most once in all arrangements (in this case exactly once).
here they are:
1–2–3–4–5–6–7–-1
1–3–5–7–2–4–6–-1
1–4–7–3–6–2–5–-1
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