Professor Plum has a dinner party with n² guests. Her guests are seated at n tables, with n guests at each table. After each dish, a new seating arrangement is made so that *every* guest sits down at a table with guests they have not already shared a table with.

Can *every* guest share a table with every other guest during the dinner?
If not, what requirements do we have on n to allow all the guests to share tables with each other?

(The poster is the original author of this problem.)

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## Comments

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TopNewestCould you, please, mention the resource (the author) of the problem. I liked it very much.

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This is a problem I came up with myself, so thanks! (I'll add a short note on that in the original problem.) I'm sure there are similar problems out there, but so far I haven't seen any.

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I know that every guest can dine with every other guest if n is a prime number (see version 1 of the problem for comments on this). I've also been told that a similar approach can be used if n is a power of a prime number (eg. n=2^3).

I don't think every guest can meet all other guests if n has more than one prime factor (eg. n=6), but I am not sure.

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