Math brain teaser problem?
Four high schools - Central, Madison, Lakeside and Western - Competed in an all-city soccer tournament. Each school played each of the others once. In each game, two points were at stake: the winner received two points, but if the two teams tied, each received one point.
At the end of the tournament, the total was: Madison, 5 points; Central, 3 points; Western, 3 points; Lakeside, 1 point.
In the entire tournament, a total of THIRTEEN goals were scored, seven of them by Central High alone; Western High scored no goals at all.
In the game between Central and Lakeside, Central won four goals to one.
Can you fill in the scores to the other five games?
Central-Madison
Central-Lakeside 4-1
Central-Western
Madison-Lakeside
Madison-Western
Lakeside-Western
Comments
Central-Madison 3-4
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside 1-0
Madison-Western 0-0
Lakeside-Western 0-0
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1. Western scored no goals, but gained three points. Each of its three games must have resulted in a 0-0 draw. So ...
Central-Madison
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside
Madison-Western 0-0
Lakeside-Western 0-0
2. Central scored 7 goals ...
Central-Madison 3-
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside
Madison-Western 0-0
Lakeside-Western 0-0
... but lost the game against Madison ...
Central-Madison 3-(4, 5, 6...?)
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside
Madison-Western 0-0
Lakeside-Western 0-0
3. Only 13 goals were scored. If the Central-Madison game result was 3-4, that would leave one other "left over" goal ... which fits nicely into the Madison-Lakeside game.
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If Western scored 0 goals in the whole tournament, then all 3 points must come from 0-0 ties, so
Central-Madison
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside
Madison-Western 0-0
Lakeside-Western 0-0
Central's other 3 goals must be in their first game, but they must not have won because they already have 3 points, so
Central-Madison 3-(>3)
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside
Madison-Western 0-0
Lakeside-Western 0-0
That gives us at least 12 goals. We know that Madison must have beaten Lakeside, so there's the last goal.
Central-Madison 3-4
Central-Lakeside 4-1
Central-Western 0-0
Madison-Lakeside 1-0
Madison-Western 0-0
Lakeside-Western 0-0