# Social squares of prime size

by Richard A. DeVenezia, back to round robin

I call this a social square, and first encountered the problem from an entry in the guestbook. If you know a more mathematical or formal name, please let me know.

## Problem

Schedule M x M players to play in M+1 rounds of M teams of M players each, so that every player is paired with every other player only once.

N = M x M. What if you have N players and N is slightly less than a square N*?
Plan for N*=N+G players, where G is the number of ghost players. Some teams will have more players than others.

What if you have N players and N is slightly more than a square N*?
Plan for N*=N-X players, where X is the number of excess players. In the first round, put X players on standby. In subsequent rounds randomly replace X players with those on standby. Some players will play more (or less) than other players.

## Solution

This culmination of a wandering study of patterns arose from an extravagant algorithm.

Round 0 is 0..N-1 listed column-wise.
Round M is Round 0 transposed.
Round k is Round k-1 after each column j has been rotated by j positions.

PlayerId ( r, t, p ) = p x m + ( t + p x r ) % m
PlayerId ( m, t, p ) = t x m + ( p )
r = round 0..m-1,
t = team 0..m-1
p = player 0..m-1
All row-wise pairings of PlayerIds are distinct over r,t,p only when m is prime.

There may be generators for non-prime m, but I have not found them. The may be proofs indicating which non-prime m are unplannable, but I have not gone looking for them.

Enter a prime number: