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 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 |