In how many ways can the couples sit? $4$ married couples are to be seated on a circular table with $8$ identical seats. In how many ways can they be seated so that 
(i) males and females sit alternately and
(ii) no husband sits adjacent to his wife 
There are so many cases that I get confused in between (Rather, I am starting to believe that writing down each case explicitly is easier than solving it using factorials).
But then what if the question states $5$ couples instead of $4$. Can it be solved in general for $n$ couples too? Help.
$Note$- Conditions (i) and (ii) should hold simultaneously.
 A: For this particular problem there is an easy way to count (but right now I don't see how to generalise this to more than $4$ couples). One can first seat the women in $4$ alternating seats. Assuming you mean to identify rotationally symmetric arrangements this can be done in $3!=6$ ways: the first women serves as reference and her seat can be numbered 0, and seating the three other women is given by a bijection to the seats $2,4,6$. (If you also want to identify reflection symmetry, divide by $2$.)
Now to seat the men, there are two options for the husband of the lady in seat $0$, namely seats $3$ and $5$. But when this is done, the arrangement is fixed. Supposing he took seat $3$, then this seat is no longer available for the husband of the lady in seat $6$, who then must go to seat $1$; then the husband of the lady in seat $4$ must go to seat $7$, and the remaining husband to seat $5$. In case the first husband took seat $5$, the situation is similarly fixed, reasoning in the opposite direction. So in all there are $6\times 2=12$ solutions.
Added I finally found out that for $n$ couples the number is given (up to a factor $(n-1)!$ for seating the women first) by A000197 in OEIS. Presumably you can find useful things in the comments and formulas there.
A: For part (i), we start with a male. There are $4$ ways to chose the first person. The person to his left is a female, so we choose one of the four females. Again, there are $4$ ways to do this. Now we have three males from which to choose. Do you see the logic? This problem is equivalent to the counting number of Hamiltonian Cycles in $K_{n, n}$. 
For part (ii), start by choosing a starting person. There are $8$ ways to do this. There are $6 * 5$ ways to choose a person to the left ($p_{2}$) and a person to the right ($p_{3}$) of $p_{1}$, as we exclude $p_{1}$'s spouse. So now we have $6$ people left. Now how many ways can we seat someone to the left of $p_{2}$? Think along this logic.
Another good way to look at (ii) is as a derangement. We have four husbands and four wives. We consider the set $H = \{1, 2, 3, 4\}$ to represent husbands. In how many ways can we permute $H$ such that $H(i) \neq i$? There is an inclusion exclusion argument for this. Note the subfactorial approximation you may find by googling is just that- an approximation.
Edit: I did not realize both conditions had to hold. 
This is still a bipartite Hamiltonian Circuit problem. Start by choosing a husband $h_{1}$. There are $4$ ways to do this. Then look at the wives set. We must have a wife on either side of the chosen husband that is different from this husband's spouse. Call these wives $w_{2}, w_{3}$. There are $\binom{3}{2}$ ways to accomplish this. We then look at a wife chosen, $w_{2}$. She has $2$ husbands to pick from other than her own. This new husband $h_{3}$ has two wives to choose from. Do you see the logic here? Can you continue from this point?
A: I shall firstly assume numbered seats, and count ways in which at least 1 couple is together, at least 2 couples are together, etc and then apply inclusion-exclusion.
The items, in sequence, represent
a) Choose couples: 4c1, 4c2, 4c3, 4c4
b) Place them: 8/7 *7c1, 8/6 *6c2, 8/5 *5c3, 8/4 *4c4
[with 2 couples together, say, there are 2+4singles = 6 "blocks" available so the couples can be placed in 6c2 ways, multiplied by 8/6 because there are only 6 "blocks" against 8 positions]
c) Arrange them: 1!, 2!, 3!, 4!
d) Flip spouses: 2¹, 2², 2³, 2⁴ 
e) Permute rest: 6!, 4!, 2!, 0!
Applying inclusion-exclusion, 
number of ways = 
8! - 4*8*1*2*6! + 6*20*2*4*4! -4*16*3!*8*2! +1*2*4!*16*1 = 11,904
Now we divide by 8 to take care of "identical seats" to get ans = 1488
You can now generalise this to take care of any number of couples ! 
edit
I erred by taking only condition (ii) into account, and solved instead another version of the ménage problem !
But re Marc's observation : is there a way to generalise beyond 4 for the correct problem ?
