### Football Board

Source: The Hidden Mathematics of Sport

Problems:

Football tables have been the basis of many a brainteaser over the years. These two puzzles ask you to work out what the scores were in all matches played so far this season.

Puzzle 1: Each team played the others once, what were the scores in each match (2 points for a win, 1 for a draw)?

 Played Goals for Goals against Points United 2 1 0 3 City 2 2 1 2 Albion 2 0 2 1

Puzzle 2: The league table below got smudged in the rain, and is only partly legible. Eventually each team will play the others once, but the tournament isn't over yet. Can you find all the results?

 Played Won Lost Drawn Goals for Goals against Points Athletic 3 2 * * 4 4 * Rovers * 1 * 0 3 0 * Town * * * 0 1 1 * Wanderers * * * * * * *

Update (27 Aug 2011):
Solution posted by Gautam Kamath (EE, Senior Undergraduate, IIT Bombay) in comments!

1. Puzzle 1:
United vs City : 1-0
United vs Albion : 0-0
City vs Albion : 2-0

Puzzle 2:
Athletic vs Rovers : 0-3
Athletic vs Town : 1-0
Athletic vs Wanderers : 3-1
Town vs Wanderers : 1-0
(because there is only 1 goal against Town which was scored by Athletic as Athletic won that game)

This puzzle seems to be very similar to Sudoku.

2. Correct! So, the second table becomes

Athletic: Played 3, Won 2, Lost 1, Goals For 4, Goals Against 4, Points 4

Rovers: Played 1, Won 1, Lost 0, Goals For 3, Goals Against 0, Points 2

Town: Played 2, Won 1, Lost 1, Goals For 1, Goals Against 1, Points 2

Wanderers: Played 2, Won 0, Lost 2, Goals For 1, Goals Against 4, Points 0

3. Yeah, pretty similar to Sudoku!