If 9 people are seated in a row of 12 chairs, then at least one consecutive set of 3 chairs is occupied by people.
(In reply to re(2): Sparseness .....spoiler
by ed bottemiller)
It was not obvious that you refer to the "sparsest set ".
No doubt, you fully ubnderstood the problem, but Justins explanation sounds much more rigid.
Anyway , we all know that we don't have to list all the possible partitions of 9 into 4 or less integers.