How to accommodate n couples at a round table so that everyone sits at the side of own companion and the other side of same sex？ [closed]

Without loss of generality, let the people be Mr. and Mrs. $A$, Mr. and Mrs. $B$, etc...

Let us avoid the challenges that the roundness of the table cause by designating Mrs. $A$ as special and allowing her the freedom to sit anywhere at the table she wishes and just referring to how people sit in relation to her. It is unnecessary to consider how many options she had here as all choices are in effect going to lead to effectively the "same" outcome otherwise.

Now that Mrs. $A$ is sitting, we are told that Mr. $A$ must sit next to her. Decide whether this will occur on her left or on her right. ($2$ choices)

Continuing in that direction (whichever it was), choose which man will set next to Mr. $A$. ($5$ choices)

Whichever man that happened to be, let his wife sit on the other side of him ($1$ choice). Then, choose which other woman will sit next to her ($4$ choices)

Continue in this fashion until all are seated.

Note in particular that were we to continue, we do have the other person seated next to Mrs. $A$ will indeed end up being another woman.

If there were an odd number of couples in total, is that still going to be the case?

Apply rule of product and conclude.