Given the data below, who will be receiving what gift?

  1. Mr. Frantz will not get the socks unless Mr. Pleacher gets the tie.
  2. Mr. Frantz will not get the vest unless Mr. Vogel gets the socks.
  3. Mr. Frantz will not get the tie unless Mr. Pleacher gets the vest.
  4. Mr. Vogel will not get the socks unless Mr. Frantz gets the tie.
  5. Mr. Pleacher will not get the vest unless Mr. Vogel gets the tie.


Answers:
Mr. Frantz gets the socks.
Mr. Pleacher gets the tie.
Mr. Vogel gets the vest.


Explanation:
Begin by rewriting each of the five clues in equivalent "IF... THEN... form."
  1. If Mr. Pleacher gets the tie, then Mr. Frantz will get the socks.
  2. If Mr. Vogel gets the socks, Mr. Frantz will get the vest.
  3. If Mr. Pleacher gets the vest, Mr. Frantz will get the tie.
  4. If Mr. Frantz gets the tie, Mr. Vogel will get the socks.
  5. If Mr. Vogel gets the tie, Mr. Pleacher will get the vest.


Use INDIRECT REASONING to determine who gets what gift.   In an indirect proof, you assume the opposite of what you are trying to prove, and then find a contradiction.

Since we do not know the answers, we will take each statement as true, then look for a contradiction.   Let's begin with statement #5.
If Mr. Vogel gets the tie, Mr. Pleacher will get the vest.   That means that Mr. Frantz gets the socks.
However, statement 3 says that If Mr. Pleacher gets the vest, Mr. Frantz will get the tie.
But we know Mr. Vogel gets the tie.
So, this can't be true, and the statement #5 does not work.

Now assume that statement #4 is true:
If Mr. Frantz gets the tie, Mr. Vogel will get the socks.
That means Mr. Pleacher gets the vest.
However, statement #2 says that If Mr. Vogel gets the socks, Mr. Frantz will get the vest.
But Mr. Pleacher must get the vest, so again, this is a contradiction.
So statement #4 cannot be true.

Now assume that statement #3 is true:
If Mr. Pleacher gets the vest, Mr. Frantz will get the tie.
That means that Mr. Vogel will get the socks.
But statement 2 says that If Mr. Vogel gets the socks, Mr. Frantz will get the vest.
Again this is a contradiction since Mr. Pleacher got the vest.
So statement #3 cannot be true.

Now assume that statement #2 is true:
If Mr. Vogel gets the socks, Mr. Frantz will get the vest.
That means that Mr. Pleacher gets the tie.
But statement #1 says that If Mr. Pleacher gets the tie, then Mr. Frantz will get the socks.
This is a contradiction since we know that Mr. Vogel got the socks.
So statement #2 cannot be true.   Therefore statement #1 must be true unless it is a bad problem.

Assume statement #1 is true:
If Mr. Pleacher gets the tie, then Mr. Frantz will get the socks.
That means that Mr. Vogel gets the vest.
None of the other four statements contradict that, so this is the answer.