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:

round table.jpg

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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