The Sportsville Teams

Date: 10/30/97 at 02:00:16
From: isak
Subject: Logic

Our teacher gives us these kinds of logic problems each  week. They 
have to do with combining the clues, and I think they are really hard. 
Sometimes I solve some of a problem, but I am seldom able to finish 
all of it. Are there straightforward methods for solving logic 
problems like this one?   

Sportsville teams. 

There are four stadiums in Sportsville: Memorial, the Coliseum, 
Central, and All Saints. These are the homegrounds for football, 
soccer, baseball, tennis, and basketball teams. Two teams share the 
same stadium. The five teams are the Blazers, the Fireballs, the 
Streaks, the Flames, and the Demons. 

From the clues given, try to determine the NICKNAME of each sports 
team and the STADIUMS at which they play: 

1) Neither the Demons nor the team that plays at Central must share 
   its stadium but the Flames must.
 
2) The football team doesn`t play at Central stadium and it shares its 
   stadium with the Streaks. 

3) The basketball team, the baseball team, and the Fireballs do not 
   share their stadiums. 

4) The soccer team is not called the Fireballs and doesn`t play at All 
   Saints. 

5) The tennis team plays at Memorial stadium but the baseball team 
   does not play at All Saints.

Date: 10/30/97 at 11:13:13
From: Doctor Rob
Subject: Re: Logic

To do problems like these, I make a diagram to help me keep track of 
what is known.  

In this case there are four attributes: stadium, sport, nickname, and 
sharing. This means that my diagram will have 1 + 2 + 3 = 4*(4-1)/2 
= 6 parts.  Each part will be a rectangular array.

The first part will be stadium versus sport, so it will have four rows 
and five columns. Each row will be labeled with the name of a stadium, 
and each column with the name of a sport. In any cell in the array, I 
put a 1 if that sport is played in that stadium, and a 0 otherwise.

The second part will be stadium versus nickname, so it will also have 
four rows and five columns. Each row will be labeled with the name of 
the stadium, and each column with the nickname of a team. In any cell 
in the array, I put a 1 if that team plays in that stadium, and a 0 
otherwise.

The third part will be stadium versus sharing, so it will be 4-by-1.

The fourth part will be sport versus nickname, so it will be 5-by-5.

The fifth part will be sport versus sharing, so it will be 5-by-1.

The sixth part will be nickname versus sharing, so it will be 5-by-1.

In each row or column of each array there will be either a single 1 
and the rest 0s, because each sport is played in only one stadium, and 
so on; or else there will be exactly two 1s and the rest 0s, because 
one stadium hosts two teams, and so on. There should be exactly five 
1s in the 4-by-5 and 5-by-5 arrays, two 1s in the 5-by-1 arrays, and 
one 1 in the 4-by-1 array.

Now we carefully analyze the clues for facts that they imply. I use 
the symbol "=>" to mean "implies". Each fact will allow us to put a 0 
or a 1 in one of the six arrays.

1) Neither the Demons nor the team that plays at Central must share 
   its stadium but the Flames must.
2) The football team doesn`t play at Central stadium and it shares its 
   stadium with the Streaks. 
3) The basketball team, the baseball team, and the Fireballs do not 
   share their stadiums. 
4) The soccer team is not called the Fireballs and doesn`t play at All 
   Saints. 
5) The tennis team plays at Memorial stadium but the baseball team 
   does not play at All Saints.

5) =>
F1: The tennis team plays at Memorial stadium.
F2: The tennis team does not play at the Coliseum.
F3: The tennis team does not play at Central.
F4: The tennis team does not play at All Saints.
F5: The baseball team does not play at All Saints.

4) =>
F6: The soccer team is not the Fireballs.
F7: The soccer team does not play at All Saints.

1) =>
F8: The Demons do not share their stadium.
F9: Central is not the shared stadium.
F10: The Flames share their stadium.

2) =>
F11: The football team doesn't play at Central.
F12: The football team is not the Streaks.
F13: The football team shares its stadium.
F14: The Streaks share their stadium.
F9 above.

3) =>
F15: The basketball team does not share its stadium.
F16: The baseball team does not share its stadium.
F17: The Fireballs do not share their stadium.
F18: F10 & F14 => The Streaks and Flames share their stadium.
F19: F10 & F14 => The Blazers do not share their stadium.
F20: F13 & F12 & F18 => The football team is the Flames.
F21: F20 => The football team is not the Blazers.
F22: F20 => The football team is not the Fireballs.
F23: F20 => The football team is not the Demons.
F24: F20 => The soccer team is not the Flames.
F25: F20 => The baseball team is not the Flames.
F26: F20 => The tennis team is not the Flames.
F27: F20 => The basketball team is not the Flames.
F28: F18 & F16 => The baseball team is not the Streaks.
F29: F18 & F15 => The basketball team is not the Streaks.

Now if the Streaks played at All Saints, by F4, F5, and F7, they could 
not be the tennis, baseball, or soccer teams, and by F29 and F12, they 
are not the basketball or football teams.  This is a contradiction, so

F30: The Streaks do not play at All Saints.
F31: F30 & F18 => All Saints is not the shared stadium.
F32: F31 & F18 => The Flames do not play at All Saints.
F33: F18 & F9 => The Flames do not play at Central.
F33: F18 & F9 => The Streaks do not play at Central.
F34: F32 & F20 => The football team does not play at All Saints.
F35: F34 & F4 & F5 & F7 => The basketball team plays at 
All Saints.
F36: F35 => The basketball team does not play at Central.
F37: F35 => The basketball team does not play at the Coliseum.
F38: F35 => The basketball team does not play at Memorial.

Questions:
Does 1) => The Demons do not play at Central?
Does 3) => The Fireballs do not play basketball or baseball?

These are questions of semantics, not of mathematics or logic, I 
think. They need to be clarified with your teacher. In this case, the 
answer is not unique unless this is what is meant, so I suppose we can 
go ahead and assume that that is what is meant.

1) =>
F39: The Demons do not play at Central.
3) =>
F40: The Fireballs do not play basketball.
F41: The Fireballs do not play baseball.

These last three facts imply many further facts, starting with the
Fireballs playing tennis (F6, F22, F40, F41).

At some point we may have to make an assumption.  Assume one of a 
small set of possibilities, and try to see what it implies. Call the 
assumption A1, and the consequences it implies C1, C2, ...  If the 
assumption implies a contradiction, it must be false, and the set of 
possibilities is reduced in size. Then discard A1 and its 
consequences, and continue.  This is what we did after F29 above.

This is the way that these logic puzzles are worked out. Usually they
are constructed so that there is only one solution, and you are more 
or less forced to discover it by this method.

With this start, you should be able to finish the problem.

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

Date: 11/03/97 at 13:03:15
From: Sky
Subject: Re: Logic

Thank you so much for the help with the logic problem!  I was able to 
sort it out by keeping track of the information the way you taught me. 
I am sharing this help with some friends.

Again, thanks!
Isak