# Sunday Times Teaser 2779 – New Year Party

*by Nick MacKinnon*

#### Published: 27 December 2015 (link)

We have a game planned for our forthcoming New Year party. Each person there will write their name on a slip of paper and the slips will be shuffled and one given to each person. If anyone gets their own slip, then all the slips will be collected up and we shall start again. When everyone has been given a name different from their own, each person will use their right hand to hold the left hand of the person named on their slip. We hope that everyone will then be forming one circle ready to sing Auld Lang Syne — but there’s a slightly less than evens chance of this happening.

How many people will there be at the party?

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We can calculate d(n) = the number of derangements of n elements, and c(n) = the number of cyclic permutations of n elements, by recursive formula, or, as we can build them up from the previous terms as we consider increasing n.