Last wee I posted this puzzle…..

Four heterosexual men and four heterosexual women are shipwrecked on a desert island.

Eventually each one falls in love with one another, and is loved by one person.  John falls in love with a girl who is, unfortunately, in love with Jim.  Arthur loves a girl who loves the man who loves Ellen.  Mary is loved by the man who is loved by the girl who is loved by Bruce.  Gloria hates Bruce and is hated by the man whom Hazel loves.
Who loves Arthur?

If you have not tried to solve it, have a go now.  For everyone else, the answer is after the break.

Gloria loves Arthur.  Did you solve it?  How?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.


  1. I wrote a ruby script which looped through all possibilities (24 ways of organising the men, and 24 for the women, so 576), and checked to see if the constraints were followed.

    Got 2 answers, and Gloria loved Arthur in both.

    Answer 1:
    john loves mary
    jim loves gloria
    arthur loves hazel
    bruce loves ellen

    ellen loves john
    mary loves jim
    hazel loves bruce
    gloria loves arthur

    Answer 2:
    john loves ellen
    jim loves gloria
    arthur loves hazel
    bruce loves mary

    ellen loves jim
    mary loves bruce
    hazel loves john
    gloria loves arthur

    1. I originally got only the first answer, but had earlier on realized that the second and third clues could each potentially imply mutual love (if one is willing to accept that Richard is being tricky). After everyone else pointed out that two answers were possible, I went back and reconsidered that hypothesis and came up with your second answer. Probably Richard only intended the first answer, but since the scope of the question is so much more narrow it hardly matters.

  2. It took me a few minutes to solve. It is clear that no two people love each other, so we need to piece together the loop, alternating men and women. And we have five clues:

    1. John > woman > Jim
    2. Arthur > woman > man > Ellen
    3. Bruce > woman > man > Mary
    4. Gloria > Not Bruce
    5. Hazel > man > Not Gloria

    From clue 1, Jim must follow John, so we can have two options:
    Bruce > woman > Arthur > woman > John > woman > Jim > woman
    Arthur > woman > Bruce > woman > John > woman > Jim > woman

    Substituting in clues 2 and 3, we then have:
    Bruce > woman > Arthur > Mary > John > Ellen > Jim > woman
    Arthur > woman > Bruce > Ellen > John > Mary > Jim > woman

    According to clue 4, we place Gloria not before Bruce. Then Hazel goes in the last remaining space:
    Bruce > Gloria > Arthur > Mary > John > Ellen > Jim > Hazel
    Arthur > Hazel > Bruce > Ellen > John > Mary > Jim > Gloria

    The first of which violates clue 5. So the true order is:
    Arthur > Hazel > Bruce > Ellen > John > Mary > Jim > Gloria

    Gloria loves Arthur. Q.E.D.

    1. Brilliantly explained. I have to admit I got completely lost on this one but I follow your logic completely. Thanks.

    2. “It is clear that no two people love each other, so we need to piece together the loop”

      Well, no – you might have had…
      – One couple who love each other, and a loop of six
      – Two couples and a loop of four
      – No couples, but two loops of four
      …or many other combinations.

      Your reasoning works for a single loop of eight, but doesn’t show it’s the only way. As it turns out there are two arrangements that fit the clues:
      – A loop of eight, as in your answer
      – A couple and a loop of six, as in Gib’s answer

      But either way the answer is Gloria.

  3. Is there a “parallel thinking” type of answer, or is it purely mathematics ?
    If it’s a math quizz… what is the point of publishing it on your blog Richard ?

  4. When I shared this with friends, I changed the wording of the question to make it unambiguously have only one answer. I also swapped Jim and John around so I could tell if someone copied and pasted an answer from google 😉

    Four men and four women are shipwrecked on a desert island. Each one falls in love with one another, and is loved by one person. No one loves the person who loves them, and everyone is loved by someone of the opposite gender. Jim falls in love with a woman who is in love with John. Arthur loves a woman who loves the man who loves Ellen. Mary is loved by the man who is loved by the woman who is loved by Bruce. Gloria does not love Bruce and is not loved by the man whom Hazel loves. Who loves Arthur?

    I worked it out using excel (in a similar way to how some people worked it out using strips of paper)
    Here is the image of the final answer:

  5. Oh yeah and I also changed all the instances of “girl” to “woman” in my example above. It seemed a little creepy otherwise :p

    The other difference was changing “hate” to “not love,” since some people seemed to feel that you could love and hate a person simultaneously.

  6. I guessed, as I hate these kinds of puzzle and couldn’t be bothered to write out all the possibilities. It was a 1 in 4 chance I’d get it right by picking a name at random.

    I got the wrong name

  7. I got the wrong name. I assumed most of this was a bunch of red herrings so focused on one sentence: ‘Arthur loves a girl who loves a man who loves Ellen’. If the girl Arthur loves is Ellen, and the man Ellen loves is Arthur, it all works out fine without needing to worry about anyone else.

  8. Wow, I got Gloria, yay! I worked it out by having the list of guys names opposite the list of girls names, listed in order as they appeared in the puzzle. I then arrowed the guys across to the girls and then kept moving the arrows down/up one position at a time until I could follow on to an answer that worked.

  9. I considered the “hate” clue to be irrelevant sine, logically, that does not exclude “loving” the same person. And these are supposed to be logic puzzles, right?

    1. That’s what I said in the comments on the original. If ‘hates’ doesn’t exclude ‘loves’ then several answers are possible. And if the intention is to exclude love, then the puzzle should just have said ‘X does not love Y’.

  10. The next puzzle is how does the population develop if none of their feelings are mutual, do they have turkey basters on their island?

  11. I wrote down all the conditions posted in the Friday Puzzle and I started looking where there would be an overlap. (For example, John loves a woman who loves Jim, and Hazel loves a man who does not love Gloria… so I tried fitting Hazel in the first condition, so the man who loves Hazel was Jim.) I kept on trying until I finally had a result that matched all the conditions, because in all the other ones, Gloria ended up in a place she couldn’t be thanks to her hating Bruce. That’s how I figured it out.

  12. I got one other solution, i’ve not read hear till now:

    The clues were:
    H->♂ / G / B
    I hope the symbols for (fe)male won’t vanish when I post this.
    With -> I mean “who loves”
    With / I mean “who hates”
    And I use abbreviations with two letters for men, and with one for women.

    My thought was: Why should be there just one big circle like this:

    ♂->♀->♂->♀->♂->♀->♂->♀ (who loves the first man of course)

    So I tried a few other combinations.
    Two circles of four pearsons didn’t work.
    ♂->♀->♂-> ♀
    You always get one circle with Jo and Ji in it (Because of the first clue). So in the second circle must be Ar and Br and because of the second and third clue in this circle are also M and E.
    So for the first circle stay two possibilities:
    H->Jo->G->Ji or H->Ji->G->Jo. But both are against the fourth clue.

    Ok, next thougt: Why shoudn’t we gat one happy pair? for example:
    Br->M->Br->M, that wouldn’t hurt the clue.
    so there could be one pair and one circle of six persons:

    When you try this with ArE you always get problems with the fourth clue (belive me or try it)
    But with BrM its getting better:
    Because of the first clue we have:
    so the missing man must be Ar. And the position of Ar gives because of the second clue also the position of E:
    The two missing girs are G and H. Because of the last clue it can’t be H->Ar->G, the man who is loved by H must hate G. But G->Ar->H is possible. and that it is:

  13. Not interesting. Clearly it could be solved by exhaustive search, as Gib presented, which would be tedious and time-consuming, so I couldn’t be bothered actually executing it. I supposed that there would likely be some quick trick, but couldn’t see one. Turns out that on this occasion I hadn’t overlooked anything.

  14. I wrote down each clue on a strip of paper (eg John > female > Jim) with equal spacing between protagonists. I then aligned these horizontally as I decided which unknown female/male related to which name. It was simple using this method.

    1. I can sympathize with your thoughts. Most weeks I give an effort to find out the correct answer, but this puzzle appeared too dull and time consuming to gain my attention.

  15. I started the same way as Nick and used the same notation, too. In particular, flipping around the clues that say “loved by…” so that the clues I re-wrote are only listed as X loves Y.

    Pondering the list, I couldn’t do direct substitutions to get anywhere, so I looked for something that would cut down my search space, so I *asserted* with rule 5 on rule 3 that:

    Hazel -> Bruce -> (not Gloria) -> _____ -> Mary

    Now we’re cooking. There are four women and Hazel, Mary and not-Gloria are accounted for, leaving only Ellen eligible for the not-Gloria slot.

    Hazel -> Bruce -> Ellen -> _____ -> Mary

    Looking at where the men come in with the girl -> boy -> girl, we now substitute rule 1:

    Hazel -> Bruce -> Ellen -> John -> Mary -> Jim

    Now slot in rule 2, keying off Elen’s position (filling it with Hazel and Bruce):

    Arthur -> Hazel -> Bruce -> Ellen -> John -> Mary -> Jim

    This validates with rule 4 (Gloria -> not Bruce).

    We’ve run out of rules, and we still have one person left. We add Gloria to the beginning (and/or end):

    Gloria -> Arthur -> Hazel -> Bruce -> Ellen -> John -> Mary -> Jim

    Run through the rules again, we have no violations. Gloria loves Arthur.

    I narrowed my search to give me a head start and was lucky with my first assertion, taking 15 minutes.

  16. John > x > Jim
    Arthur > x > y > Ellen
    Bruce > x > y > Mary
    Hazel > y
    Y > not Gloria
    Gloria > not Y
    Arthur > Mary > John > Ellen
    John > Ellen > Jim
    Bruce > Gloria > Arthur > Mary
    Hazel > Bruce
    Gloria > Not Bruce
    Arthur > Mary
    John > Ellen
    Bruce > Gloria
    Jim > Hazel
    Mary > John
    Ellen > Jim
    Hazel > Bruce
    Gloria > Arthur
    John > Ellen > Jim
    Arthur > Mary > John > Ellen
    Bruce > Gloria > Arthur > Mary
    Hazel > Bruce
    Bruce > Gloria
    Gloria > Arthur

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