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Chess Tournament solution check

  1. Aug 27, 2009 #1
    Not completely sure what branch of mathematics this problem belongs in, so please move it if its in the wrong sub-forum:biggrin:. This isn't really a homework problem, but one of the weekly questions posted by the mathematics department at my school. I think I have arrived at the correct solution, but would appreciate it if someone could check it for me. I will attempt to walk you through my reasoning as well as I can. Thanks!

    Note: sorry for all the () I couldn't get the spacing to work correctly for me :frown:

    1. The problem statement, all variables and given/known data

    In a chess tournament in which each participant played the other participants exactly once, the order of finish was A first, B second, C third, D fourth, and E last. Sports Illustrated quoted B as saying "I would never have imagined that I'd be the only one who'd finish without a single loss", while player E was quoted as saying "And I'm the only one who didn't win at least one game." Based on this information, determine the outcome of each match in the tournament. Please recall the winner of a game gets 1 point, while in a tie each player receives 1/2 point. In this tournament no two players got the same number of points.

    2. Relevant equations

    20 possible combinations:

    A vs { B,C,D,E}
    B vs {A, C,D,E}
    C vs {A,B, D,E}
    D vs {A,B,C, E}
    E vs {A,B,C,D }

    Of these 20 combinations, 10 are unique.

    This is the list of the 10 matches played between the 5 chess players:

    A-B
    A-C
    A-D
    A-E
    B-C
    B-D
    B-E
    C-D
    C-E
    D-E

    After determining the matches played, I wrote out the characteristics of each player and their collection of matches:

    [A] : -highest amount of points, first place
    -at least one game lost
    -at least one game won

    : -second place
    -at least one win
    -zero losses

    [C] : -third place
    -at least one win
    -at least one loss

    [D] : -fourth place
    -at least one win
    -at least one loss

    [E] : -last place
    -at least one loss
    -zero wins


    3. The attempt at a solution

    Given the above conditions, I wrote down the possibilities for each of the 10 matches:

    1. (A-B) (B-win OR tie)
    2. (A-C) (A-win OR C-win OR tie)
    3. (A-D) (A-win OR D-win OR tie)
    4. (A-E) (A-win OR tie)
    5. (B-C) (B-win OR tie)
    6. (B-D) (B-win OR tie)
    7. (B-E) (B-win OR tie)
    8. (C-D) (C-win OR D-win OR tie)
    9. (C-E) (C-win OR tie)
    10. (D-E) (D-win or tie)

    Given the above possibilities, I wrote out tournament results that I think satisfy all constraints along with points earned each match:

    1. (A-B) (B-win) (A=0 B=1)
    2. (A-C) (A-win) (A=1 C=0)
    3. (A-D) (A-win) (A=1 D=0)
    4. (A-E) (A-win) (A=1 E=0)
    5. (B-C) (tie) (B=1/2 C=1/2)
    6. (B-D) (tie) (B=1/2 D=1/2)
    7. (B-E) (tie) (B=1/2 E=1/2)
    8. (C-D) (C-win) (C=1 D=0)
    9. (C-E) (tie) (C=1/2 E=1/2)
    10. (D-E) (D-win) (D=1 E=0)

    Tournament point totals:

    A = 3
    B = 2.5
    C = 2
    D = 1.5
    E = 1

    Sorry for the length of the post, any input is appreciated!
     
  2. jcsd
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