September 28, 1998
Problem of the Month

Soccer Stumper
Proposed by David L. Book,
in The Washington Post.

In the first round of this year's World Cup, the teams in Group G finished as follows:

   TEAM        GF    GA   Points
  ----------  ----  ----  ------
  Romania       4     2      7
  England       5     2      6
  Colombia      1     3      3
  Tunisia       1     4      1
  

GF stands for "Goals For" and GA stands for "Goals Against."

A victory earns three points and a tie one; a loss earns no points.

Each team played the other three once.

What were the scores of all the matches?

NO fair looking them up!!

Solution to the Problem:

The only way to total 7 points is by having 2 wins and a tie; the only way to total 6 points is by having 2 wins and a loss; etc; so it is easy to determine the outcome of each match.

Since all matches in this group involved little scoring, the fastest way to determine the number of goals is by trial and error.   But if you want to do it mathematically, you can construct a 4 x 4 matrix and solve for the 12 unknowns:

       Rom Eng Col Tun
   Rom  -   A   B   C   4
   Eng  D   -   E   F   5
   Col  G   H   -   I   1
   Tun  J   K   L   -   1
        2   2   3   4
    

Romania defeated England, 2-1.
Romania defeated Colombia 1-0.
Romania tied Tunisia, 1-1.
England defeated Colombia, 2-0.
England defeated Tunisia, 2-0.
Colombia defeated Tunisia, 1-0.

Correctly solved by:

1. Kaveh Sadeghzadeh	College of William & Mary, Williamsburg, VA
2. Barrett Waybright	Winchester, VA
3. Darren Rennie	Winchester, VA
4. John Jacobs	Brown University, Providence, RI
5. Michael Leatherman	Old Dominion University, Norfolk, VA
6. James Puckett II	Old Dominion University, Norfolk, VA
7. Jay O'Donnell	Winchester, VA
8. Paul Carpenter	Winchester, VA

Send any comments or questions to: David Pleacher