# Husbands and Wives around a Circular Table

Transcription from this image. Some of the list formatting has been slightly altered to make it play nicer with Markdown.

Eight persons, A through H, comprising four married couples, sit around a circular table which had eight equally spaced seats around it. The following information is known about their seating arrangement:

1. No two wives are sitting adjacent to each other and no couple is sitting adjacent to each other.
2. E, who is the husband of F, is sitting opposite B, while A, who is sitting to the left of G, is the wife of someone other than B.
3. C is sitting adjacent to a female.

Questions start here.

1. If it is known that A is sitting opposite H, then interchanging the seats of which of the following pairs of persons, in that order, will result in all the husbands sitting next to their respective wives?
1. C and G
2. A and H
3. B and E
4. G and E
5. A and F

(A) 1 and then 2
(B) 3 and then 4
(C) 1 and then 3
(D) 4 and then 5

1. If F is sitting opposite D, who is the wife of G?

(A) A
(B) D
(C) H
(D) C

1. If B is sitting next to the wife of G, who is sitting to the right of E?

(A) Wife of C
(B) Wife of A
(C) Husband of A
(D) Wife of B

1. Which of the following statements is definitely true?

(A) G is sitting to the immediate left of C.
(B) The wife of B is sitting to the immediate right of F.
(C) The wife of C is sitting opposite the wife of G.
(D) F is sitting two places away from G.

Source : time.com

What we know from the rules:

I will preface that I believe this question assumes that all marriages are between a wife and a husband.

The first clue tells us that no two wives are sitting adjacent to each other, this means that the table must alternate between wife, husband, wife, husband, etc. To prove this to yourself, try make two husbands sit next to each other without two wives sitting next to each other. This also means that adjacent people are different genders and opposite people are the same gender. From the first three rules combined we can entirely deduce that A, D, F, and H are wives and B, C, E, and G are husbands.

We also know that no couples are sitting next to each other, let's label the couples arbitrarily as W, X, Y, and Z in no particular order. If we have the seats as FMFMFMFM, then the only way that the couples can be seated is WXYZXWZY and the rotations of this like WZYWXYZX (this will be useful later).

Let's label the 8 seats 1 through 8 going clockwise around the table and make it so that wives will be on the even seats and husbands on the odd seats. We will also make E sit at seat 1. Now we can put down who could potentially sit at each seat:

• 1: E
• 2: A, D, H
• 3: C, G
• 4: D, F, H
• 5: B
• 6: A, D, F, H
• 7: C, G
• 8: D, H

In addition we can also put down who are the potential spouses of each person:

• A: C
• B: D, H
• C: A
• D: B, G
• E: F
• F: E
• G: D, H
• H: B, G

The above lists are just by going through each rule and crossing out everything that is no longer valid, since it is quite lengthy to explain each step I'll leave this as an exercise to you. Since this is enough to get you going, I'll put the answers to the questions in spoilers:

Question 1:

if H were to sit in seats 4 or 8, then A would have to be in 8 or 4 respectively but since A can't sit there, we know that H can't sit in 8 or 4.
Removing H from seats 4 and 8 shows us that it is D who sits in seat 8.
And since we know that D and H can't be in 4, then we know that F sits in seat 4

The next thing we know now is what rotation of the couple pairings we are dealing with. We are dealing with WZYWXYZX (notice that 1 and 4 are both W because that is where E and F are) and we can work out that B and D are married (they are in seats 5 and 8) and we also know now that G and H are married.

We don't actually need to know (and we can't determine) which seats belong to A, C, G, and H to answer this question. If we switch C and G firstly then this will make sure that A and C are sitting next to each other and that G and H are sitting next to each other. Then if we switch B and E the the other two couples will be sitting next to each other. This means the answer to question 1 is (C) I and then III.

Question 2:

In the last answer, we had a solution where D and F were sitting opposite from each other, so we should expect that G is married to H. If you were working this out again from scratch, you can use the fact that D and F have to be on 8 and 4 respectively otherwise there is no valid seat for A, then find the couple pairings in WZYWXYZX. The answer to this question is (C) H

Question 3:

If we look back at our answer to Q1, if we put H in seat 6, G in seat 3, A in seat 2, and C in seat 7, then we would have recreated the scenario described by this question. In that case, to the right of E is seat 8 which had D who was the wife of B, so the answer is (D) Wife of B.

Question 4:

(A) we know is false because C can't sit to the right of anywhere that G can.
(B) F and the wife of B are both wives so they can't sit next to each other
(C) The wife of C is A and we know G is to the right of A. Lets assume a couple pairing of WXYZXWZY, A can only be seat 2 as seat 6 is taken by F. It would mean that A's spouse is in seat 5 which is already taken by seat B, so this is not a valid scenario. If we instead use a couple pairing of WZYWXYZX, then regardless of whether seat A ends up in seat 2 or 6, the opposite will contain the wife of G. This all means the statement is true. (apologies for how wordy/confusing this one probably is, not sure how to word it better)
(D) All possible places for F are exactly 1 place away from G, not 2.