## League table teasers

Submitted by mf344 on June 21, 2011*This puzzle is from the excellent book The hidden mathematics of sport by Rob Eastaway and John Haigh, reproduced here with kind permission of the authors.*

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 | * | * | * | * | * | * | * |

Both puzzles require a degree of deductive reasoning. One way to begin to tackle them is to realise that, for example, the total of the *Win* column is always the same as the total of the *Lose* column (one win for one team must create one loss for another team), while the total of the *Draw* column will be twice the number of matches that have been drawn (because every draw involves two teams). Similarly, the *Goals for* and *Goals against* columns must add up to the same total.