1996 APMO 第 3 题

The National Marriage Council wishes to invite $n$ couples to form 17 discussion groups under the following conditions:

(1) All members of a group must be of the same sex; i.e. they are either all male or all female.

(2) The difference in the size of any two groups is 0 or 1.

(3) All groups have at least 1 member.

(4) Each person must belong to one and only one group.

Find all values of $n$ , $n \leq 1996$ , for which this is possible. Justify your answer.

国家婚姻委员会希望邀请 $n$ 夫妇在以下条件下组成 17 个讨论组：

(1) 团体的所有成员必须是同一性别；即他们要么都是男性，要么都是女性。

(2)任意两个组的大小之差为0或1。

(3) 所有团体至少有1名成员。

(4) 每个人必须属于一个且仅一个团体。

查找 $n$ 、 $n \leq 1996$ 的所有可能的值。证明你的答案合理。