The seating arrangement problem, version 2

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.)

#NumberTheory #PrimePower #PrimeModuli #PrimeNumbers

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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.

Johan Falk - 6 years, 11 months ago

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

Сергей Кротов - 6 years, 11 months ago

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.

Johan Falk - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$