Four gentlemen (A, B, C, and D) went to an expensive restaurant to dine. They checked their coats, hats, gloves, and canes at the door (each of the gentlemen had one of each). But when they checked out, there was a mix up, and each of the men ended up with exactly one article of clothing (a pair of gloves is considered a single article of clothing) belonging to each one of the four.
A and B ended up with their own coats, C ended up with his own hat, and D ended up with his own gloves. A did not end up with C's cane.
State whose coat, hat, gloves, and cane each of the gentlemen ended up with.
As each man had one item from each of the men, that includes having exactly one of his own. In the cases of C and D that was his own hat and his own gloves respectively. A and B, on the other hand, each had his own coat. That means Messrs. A,B,C,D had the coats of A,B,D,C respectively--that is C's and D's coats were interchanged.
As C had his own hat, and none of A,B or D had his own, those three must have had one of the two possible derangements of their three hats: A,B,D had the respective hats of either B,D,A or D,A,B.
Likewise, as D had his own gloves, and none of A,B or C had his own, those three must have had one of the two possible derangements of their gloves: A,B,C had the respective gloves of either B,C,A or C,A,B.
Thus there are only four possibilities for the three items: coats, hats and gloves. Each then determines the fourth item: the cane. But we can see the results:
Man: ABCD ABCD ABCD ABCD
coat: ABDC ABDC ABDC ABDC
hat: BDCA BDCA DACB DACB (the two possibilities)
gloves: BCAD CABD BCAD CABD (in combination with these two possibilities)
cane: ---- DCAB CDBA ---- (based on fact of one item from each of the four)
The hyphens in two of the columns represent the fact that in those columns there was already a duplication, in the fact of one of the men having more than one item from the same man.
Since A did not end up with C's cane, A,B,C,D had in the given order, the coats of A,B,D,C; the hats of B,D,C,A; the gloves of C,A,B,D and the canes of D,C,A,B.
Posted by Charlie
on 2003-09-02 15:02:20