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### Football Logic: Who Defeated Whom...?

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Date: 8/17/96 at 18:5:52
From: Anonymous
Subject: Football Logic: Who Defeated Whom...?

An elimination tournament in football was held between four
colleges - Trinity, Tufts, Temple, and Tulane.  The winners of the
first two games met in the third and final game to decide the
championship. The colors of the teams were brown, blue, red, and
purple: and the competing captains were Albie, Barry, Bill, and Ben.

Who defeated whom in the play-off: and by what score?
Who was the captain of each team?
What was the color of each team?

1. In the final game Albie's team made its only score by a touchdown
on the first play, but missed the point after the touchdown.
2. The red team lost to the Tufts in the first game.
3. Ben's team defeated Tulane 12 to 0.
4. The captain of the purple team saved his team from being scoreless
in the third game by a 40 yard field goal.
5. Ben's team did not play Trinity.
6. Barry's team lost to the undefeated team.
7. Albie did not see his former friend, the captain of the brown team.

That's the end of the puzzle.  I haven't figured much out so
please give me some help. Thanks.
```

```
Date: 8/20/96 at 16:3:37
From: Doctor Leigh
Subject: Re: Football Logic: Who Defeated Whom...?

Thank you for your question.  The best way to solve a problem like
this one is to list all of the possibilities and cross out the ones
that don't work. For this problem you would start something like this:

Trinity        Tufts         Temple        Tulane

Brown          Brown         Brown         Brown
Blue           Blue          Blue          Blue
Red            Red           Red           Red
Purple         Purple        Purple        Purple

Albie          Albie         Albie         Albie
Barry          Barry         Barry         Barry
Bill           Bill          Bill          Bill
Ben            Ben           Ben           Ben

Then you would do the same thing, but list the colors on top, then the
captains on top.  Next, using the clues cross out the names that do
not belong with each other.  For example, in the above chart for clue
#2 you would cross out red under Tufts because if red lost to Tufts
then red is not Tufts.  Also keep track of who is in the winning game,
and who lost.  Try these and see if you can come up with the answers
that I am including below.

Good Luck!!!!

-Doctor Leigh Ann,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

```

```Date: 2/28/2002 at 16:00:00
From: Doctor Ian
Subject: Re: Football Logic: Who Defeated Whom...?

Kok Leong Chin wrote to us to suggest using the following structure
to track the information in the clues:

FIRST ROUND         Game 1                      Game 2
Captain       ------      ------         ------      ------
School        ------  vs  ------         ------  vs  ------
Color         ------      ------         ------      ------
(Score)       ------      ------         ------      ------

FINALS                          Game 3
Captain                   ------      ------
School                    ------  vs  ------
Color                     ------      ------
(Score)                   ------      ------

Let's see how that works out.  From (1), we know that Albie's team was in
the final, and that it scored only 6 points:

FIRST ROUND         Game 1                     Game 2
Captain       ------      ------         ------      ------
School        ------  vs  ------         ------  vs  ------
Color         ------      ------         ------      ------
(Score)       ------      ------         ------      ------

FINALS                          Game 3
Captain                   Albie       ------
School                    ------  vs  ------
Color                     ------      ------
(Score)                   6           ------

From (4), we know that the purple team was in the finals, and
that it wasn't Albie's team, since it scored only 3 points:

FIRST ROUND         Game 1                     Game 2
Captain       ------      ------         ------      ------
School        ------  vs  ------         ------  vs  ------
Color         ------      ------         ------      ------
(Score)       ------      ------         ------      ------

FINALS                          Game 3
Captain                   Albie       ------
School                    ------  vs  ------
Color                     ------      Purple
(Score)                   6           3

From (7), we know that the brown team was on the other side of
the draw from Albie's team:

FIRST ROUND         Game 1                     Game 2
Captain       ------      ------         ------      ------
School        ------  vs  ------         ------  vs  ------
Color         ------      ------         Brown       Purple
(Score)       ------      ------         ------      ------

FINALS                          Game 3
Captain                   Albie       ------
School                    ------  vs  ------
Color                     ------      Purple
(Score)                   6           3

From (2), we know that the red team lost to Tufts in the first game.
Since Albie's team didn't lose, Albie wasn't on the red team.  So he
must have been on the blue team, Tufts:

FIRST ROUND         Game 1                     Game 2
Captain       ------      Albie          ------      ------
School        ------  vs  Tufts          ------  vs  ------
Color         Red         Blue           Brown       Purple
(Score)       ------      ------         ------      ------

FINALS                          Game 3
Captain                   Albie       ------
School                    Tufts   vs  ------
Color                     Blue        Purple
(Score)                   6           3

From (3), we know that Ben's team defeated Tulane 12-0.  This means
that Ben has to be on the purple team; and also that the purple team
defeated Tulane:

FIRST ROUND         Game 1                     Game 2
Captain       ------      Albie          ------      Ben
School        ------  vs  Tufts          Tulane  vs  ------
Color         Red         Blue           Brown       Purple
(Score)       ------      ------         0           12

FINALS                          Game 3
Captain                   Albie       ------
School                    Tufts   vs  ------
Color                     Blue        Purple
(Score)                   6           3

From (6), we know that Barry's team lost to the undefeated team.
Only the purple and red teams lost to the eventual winner, so Barry
must be captain of the red team.  And that makes Bill the captain
of the brown team:

FIRST ROUND         Game 1                     Game 2
Captain       Barry       Albie          Bill        Ben
School        ------  vs  Tufts          Tulane  vs  ------
Color         Red         Blue           Brown       Purple
(Score)       ------      ------         0           12

FINALS                          Game 3
Captain                   Albie       Ben
School                    Tufts   vs  ------
Color                     Blue        Purple
(Score)                   6           3

So far we've used (1), (2), (3), (4), (6), and (7).  What does (5)
have to tell us?  Ben's team did not play Trinity.  Now, this could
be a clever way of saying that Ben's team _is_ Trinity!  But let's
use the more conventional interpretation, that Ben's team is _not_
Trinity,  That makes the red team Trinity.  And by elimination, it
make's the purple team Temple.

FIRST ROUND         Game 1                    Game 2
Captain       Barry       Albie          Bill        Ben
School        Trinity vs  Tufts          Tulane  vs  Temple
Color         Red         Blue           Brown       Purple
(Score)       ------      ------         0           12

FINALS                          Game 3
Captain                   Albie       Ben
School                    Tufts   vs  Temple
Color                     Blue        Purple
(Score)                   6           3

Does this make sense?  Each captain, school, and color appears once, which
is good!  Let's go back and check to see if it's consistent with each clue:

1. In the final game Albie's team made its only score by a touchdown
on the first play, but missed the point after the touchdown?  Yes.

2. The red team lost to Tufts in the first game?  Yes.

3. Ben's team defeated Tulane 12 to 0?  Yes.

4. The captain of the purple team saved his team from being scoreless
in the third game by a 40 yard field goal?  Well, _someone_ did!

5. Ben's team did not play Trinity?  Right.

6. Barry's team lost to the undefeated team?  Yes.

7. Albie did not see his former friend, the captain of the brown team?
Yes.

So this is indeed a nice structure for solving the problem.  Note that
we didn't need any extra tables to track constraints, which is usually
the case in a problem like this.

-Doctor Ian,  The Math Forum
http://mathforum.org/dr.math/

```
Associated Topics:
High School Logic

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