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.
To solve this problem, we need to determine the possible values of \( n \) such that \( n \) couples can be divided into 17 discussion groups under the given conditions. Let's break down the problem step by step.

1. **Define Variables and Constraints:**
   - Let \( n \) be the number of couples, so there are \( 2n \) individuals.
   - Let \( k \) be the number of groups of males, and \( 17 - k \) be the number of groups of females.
   - Each group has either \( m \) or \( m+1 \) members.
   - The total number of males is \( n \), and the total number of females is also \( n \).

2. **Group Size Constraints:**
   - Since the difference in the size of any two groups is 0 or 1, the groups can have either \( m \) or \( m+1 \) members.
   - Let \( a \) be the number of groups with \( m \) members, and \( b \) be the number of groups with \( m+1 \) members.
   - Therefore, \( a + b = 17 \).

3. **Total Members Constraint:**
   - The total number of members in the groups of males is \( am + b(m+1) = n \).
   - Simplifying, we get \( am + bm + b = n \) or \( (a + b)m + b = n \).
   - Since \( a + b = 17 \), we have \( 17m + b = n \).

4. **Range of \( k \):**
   - We need to ensure that the groups are balanced between males and females.
   - Suppose \( k \geq 9 \) (without loss of generality, as the problem is symmetric).

5. **Case Analysis:**
   - For \( k = 11 \):
     - If \( m = 1 \), then \( 11 \times 1 + 6 \times 2 = 11 + 6 = 17 \).
     - This means \( n = 11 \) or \( n = 12 \).
   - For \( k = 10 \):
     - If \( m = 1 \), then \( 10 \times 1 + 7 \times 2 = 10 + 7 = 17 \).
     - This means \( n = 10 \) through \( n = 14 \).
     - If \( m = 2 \), then \( 10 \times 2 + 7 \times 3 = 20 + 7 = 27 \).
     - This means \( n = 20 \) or \( n = 21 \).
   - For \( k = 9 \):
     - If \( m = 1 \), then \( 9 \times 1 + 8 \times 2 = 9 + 8 = 17 \).
     - This means \( n = 9 \) through \( n = 16 \).
     - If \( m = 2 \), then \( 9 \times 2 + 8 \times 3 = 18 + 8 = 26 \).
     - This means \( n = 18 \) through \( n = 24 \).

6. **Combining Results:**
   - From the above cases, we can see that the possible values of \( n \) are:
     - \( n = 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24 \).

The final answer is \( \boxed{ n = 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24 } \).