Round Table

possible related topics  modular arithmetic, powers, sequences

Learners could begin with a smaller number of dinner guests and obtain a table of results like this:

 number of guests number of arrangements such that no one sits next to anyone else more than once 1 ? 2 1 3 1 4 1 5 2 6 2 7 3

Up to and including four people is straightforward – they have to go home once they’ve had their starter!

With five people, there are two possible arrangements:

Learners will need to think exhaustively in order to cover all possibilities.

In general, with n guests there will always be  arrangements in which no one sits next to anyone else more than once. (The square brackets indicate the floor function: rounding down to next integer below.) Learners can reason that more than  must be impossible. For instance, with 25 people, imagine being one of those people: there are only 24 possible people who can sit either side of you, meaning a maximum of 12 arrangements. To prove that  is always achievable involves mutually disjoint Hamilton cycles.

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