Three couples dine at the same restaurant every Saturday at the same table. The table is round, and the couples have agreed that
(a) husband and wife never sit next to each other at the table; and

(b) the arrangement of the six at the table is different each Saturday.

Ignoring rotations of the arrangements at the table, for how many Saturdays can these three couples go to this restaurant without repeating their seating arrangement?
To simplify, let's denote each couple by a pair of numbers, one number representing the husband and the other the wife. We have, then, the three pairs \((1,2),(3,4),(5,6)\), which cannot be neighbors. We can consider the husband 1's place at the table as fixed, since we are disregarding rotations in the arrangement around the table. Two possible arrangements are given in the following figure.

132546

164523

These arrangements are described by 132546 and 164523, always starting with 1, fixed in a position at the table, and proceeding clockwise. Thus, the problem reduces to finding all 6-digit numbers that can be written with the digits 1, 2, 3, 4, 5, and 6, where:

(a) all numbers start with the digit 1;

(b) no number can end with the digit 2;

(c) 1 and 2, 3 and 4, 5 and 6 cannot appear together.

We find the 16 numbers in the following table.

| 132546 | 132645 | 135246 | 135264 | 135426 | 136245 | 136254 | 136425 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 142536 | 142635 | 145236 | 145326 | 146235 | 146325 | 153246 | 154236 |

Therefore, the answer is 16 Saturdays.