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Feb8-12, 12:32 PM
P: 506
A little more googling shows that your problem is known as the Social Golfer Problem and has been investigated for a number of parameters. Surprisingly it appears to be the case that the solution for the original problem is unique (barring symmetry).

Note that the generalised problem ignores the 'single host' constraint of the original problem: I would guess that this is either trivial to satisfy or (if there are more meals than hosts) impossible.