Making a point

I came across this week's football mindbender in the excellent "Penguin Book of Curious and Interesting Puzzles" by David Wells.

The puzzle

To encourage teams to score more goals, a new way of awarding points has been suggested. In the new scheme 10 points are awarded for a win, 5 points for a draw and 1 point for each goal scored, whatever the result of the match. This method has been tried out on a small scale and its success can be judged by the fact that each side scored at least one goal in every match. There were only three sides playing and they will eventually play each other once. At present Rovers have been awarded 8 points, United 14 points and Borough 9 points. What are the scores in the matches that have been played so far?

The solution

The greatest number of points that could have been obtained from wins or draws alone are Rovers - 5, United - 10, Borough - 5, so only two matches have been played so far (not one as in that case one of the teams would have 0 points). Suppose that United drew two matches against Rovers and Borough. Then United's total of goals should equal Rovers + Boroughs total, but in this case United's goals total would be 4, and Rovers would be 3 and Borough's 4. Therefore Rovers vs Borough was a draw and United won one game. Either Rovers or Borough has played only one game, and since Rovers got 3 goals and Borough got 4 then Borough vs Rovers must have been a 3-3 draw and Borough scored 1 goal against United. Therefore United beat Borough 4-1.

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