Puppy love?

Logic Level 3

There are four teenage boys--Eric, Francis, George, and Henry--each of whom has a girlfriend. One day, these four boys get together and have a nice long chat about their girlfriends. However, by the time the party is over, they all have forgotten the names of the others' girlfriends (except their own girlfriend's name). Being childish as ever, all of them refused to say their girlfriend's name again.

Then the following conversation takes place:

George: "Well, my girlfriend's name isn't Erica. Does this help?"

Henry: "How would that help? Even if my girlfriend is Erica, which I won't admit or deny, I still don't know everyone's girlfriend's name. By the way, my girlfriend's name is not Georgia."

Francis: "That's interesting. My girlfriend's name does not share the same initial as mine, either."

Eric: "So Francis's girlfriend isn't Frances. Since everyone makes the assertion that their girlfriend is not of a particular name, let me join in as well: my girlfriend's name isn't Henrietta."

George: "Well, this is going nowhere. Let me give you guys another hint: my girlfriend's name is not Georgia either."

Then unbeknownst to us, someone interjects: "Finally, I know everyone's girlfriend's name."

If everyone is perfectly logical and speaks the truth, then who is the person that interjects the final statement?

Note: Each of the names of their girlfriends was spoken in the conversation above at least once.

Eric and Francis are the only possible answers Henry This is an impossible scenario It could be any of three boys: Eric, Francis and Henry

2 solutions

Saya Suka
Mar 16, 2021

The denials made among the boys can be condensed in this table below :

GFs	Erica	Frances	Georgia	Henrietta
Eric	Erica	Frances	Georgia	X
Francis	Erica	X	Georgia	Henrietta
George	X	Frances	X	Henrietta
Henry	Erica	Frances	X	Henrietta

Looking at the table, George has 2 possible girlfriends and Georgia has 2 possible boyfriends. The only way someone can figure everything out is he must be one of Georgia's possible boyfriend but actually not, so he'd knew the other possible guy must be her real boyfriend, and on top of that his real girl is also one of George possible girlfriends, so he'd definitely know the other possible girl must be George's real girlfriend. The 2 possible solutions can be seen in the tables below :

GFs	Erica	Frances	Georgia	Henrietta
Eric	X	✓	X	X
Francis	X	X	✓	X
George	X	X	X	✓
Henry	✓	X	X	X

GFs	Erica	Frances	Georgia	Henrietta
Eric	X	X	✓	X
Francis	X	X	X	✓
George	X	✓	X	X
Henry	✓	X	X	X

Emanuele Prati
Apr 17, 2019

One can be sure about the name of one’s girlfriend (I’ll call him X and her Y) if and only if a) he is sure all the guys except X are not engaged with Y or b) X’s girlfriend isn’t any of the other three girls. b) is not possible, because no one told more than two names of girls that aren’t his girlfriend, so it must be necessarily a). Precisely, one can now, other than the statements of the other guys, that Y is not his own girlfriend; to be sure about three guys who are not engaged with Y, Y must be Georgia, because other three girls have been nominated just once; so if someone knows, other than the statements of the other guys, that Georgia is neither his girlfriend, he knows who is engaged with her; this guy can be either Eric or Francis, and the guy engaged with Georgia the other one. It can also be seen that, if the girlfriend of the one who knows who is Georgia’s boyfriend is Erica, he doesn’t know all pairs: in fact, Henrietta and Frances can then be matched with either George or Henry; but if his girlfriend isn’t Erica, necessarily Henry will be engaged with Erica and George with the last girl remaining, and he will know all pairs.

Puppy love?

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