Four men and four women are shipwrecked on a deserted island. Eventually each person falls in love with one person and is loved by one person. You are given the following information:

(a)Chad loves the girl who is in love with David.
(b)Arthur loves the girl who loves the man who loves Ellen.
(c)Bruce loves the girl who loves the man who loves Mary.
(d)Gloria does not love Bruce.
(e)Helen loves a man who does not love Gloria.

There is no mutual love interest (nobody loves the person who loves them back). Nobody is homosexual or narcissistic.
Who loves who?

As there are no mutual loves, there are either 2 cycles with 4 people in each, or one full chain including all 8 people. For shorthand, I'll refer to people by their initials: A-D are men; E,M,G and H women (why couldn't Mary be called Florence, in the true puzzle tradition?)

If there are 2 cycles of 4, C and D are in one cycle because of statement (a), and that leaves A and B in the other. By (c) this would make A the man who loves M, and M would love B. By (b) B would then love E, completing this 4-cycle. That leaves G and H to fill out the 4-cycle with C and D. But that leaves (e) impossible to fulfill. Therefore there are not two cycles of 4, but rather one complete cycle of 8.

Let's represent the cycle of love by two strings--one of men and one of women, such as the following:
ABCD
EMGH
where the above placement would indicate A loves E, who loves B, who loves M, who loves C, etc. In other words the letter representing the man who loves a particular woman is directly above that woman's letter, while the man loved by that woman is up and to the right of that woman's letter.

We can always start anywhere in the cycle so we might as well put A in the upper left.

Since C loves the one who loves D, C and D have to appear directly in a row on the top row, which is therefore either ABCD or ACDB. Either way, as A loves the woman who loves the man who loves E, E must be the second letter in the lower string:
ACDB
-E--
or
ABCD
-E--

Rule (c) then places M:
ACDB
ME--
or
ABCD
-EM-

Rules (d) and (e) prevent G and H from fitting into the ACDB scenario. Then rule (d) allows only the following placement in the ABCD scenario:
ABCD
HEMG

So Arthur loves Helen, who loves Bruce, who loves Ellen, who loves Chad, who loves Mary, who loves David, who loves Gloria

Posted by Charlie on 2003-05-04 05:14:49