Eight players competed at a recent chess tournament. Knowing that:
Each player played all the others, exactly once.
Winning earns you 1 point; drawing, ½ point; and losing, 0 points.
Everybody ended with a different number of points.
The one who ended 2nd earned as many points as the four bottom players put together.
What was the result of the game between the player who ended 3rd and the player who ended 7th?
There are only 3 possible cases:
Case 1: 7th Rank defeats 3rd Rank
Case 2: 7th Rank draws with 3rd Rank
Case 3: 7th Rank loses to 3rd Rank
Before going into each of the cases, please note that:
The total points are 28. If 1st Rank has 7 points (he wins all his games),
then automatically 2nd Rank lost at least 1 game and hence has only 6 points max.
If 1st Rank has 6.5 points, then since all the players have a different score the second
Rank can only have max 6 points.
So, the max sum of the bottom 4 players is 6 points.
Now, there are 4 players at the bottom of the table and they all played with each other.
Hence 6 games were played between them and hence 6 points were shared between them. Hence
the sum of their points will at least be 6 points.
Now, this automatically eliminates case 1 and case 2 as both these cases will give the 7th
Ranked player either 1 point or 0.5 point which will make the sum of the points of the last
4 players beyond 6 points.
Hence, only case 3 is possible i.e......7th Rank loses to 3rd Rank.
-Kalam
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Posted by Kalam
on 2005-11-17 04:12:40 |
Solution
