Chess tournament with 1.5 Points Winners

Problem

Chess tournament with 1.5 Points Winners

Solution

Consider the participants as the nodes of a graph. Join two nodes with a blue edge if the game between the two players ended in draw; join two nodes with a red edge if on of the players won the game.

As the simplest example of Ramsey's theory, with the number of participants $n\ge 6,$ there is always a monochromatic triangle. In such a triangle, either all three games ended with a draw, or all three ended with a win for one of the players. In either case, there's no participant who got $1.5$ points against two other players.

For five participants it is possible to satisfy the conditions of the problem:

Chess tournament with 1.5 Points Winners, solution

In the diagram, all outer edges are similarly oriented and the direction of the vectors is, say, from the winner to the loser. All dashed, internal edges designate the games that ended in a draw.

Acknowledgment

This is problem CC124, the Contest Corner, from the Crux Mathematicorum. The commentary at VOLUME 41, NO. 5 May 2015 reports that "Originally that was question 3 on the 1999 entrance exam to the mathematical-mechanical department of the Belorussian State University. We received no solutions to this problem."

 

  1. Ramsey's Theorem
  2. Party Acquaintances
    • Chess tournament with 1.5 Points Winners
  3. Ramsey Number R(3, 3, 3)
  4. Ramsey Number R(4, 3)
  5. Ramsey Number R(5, 3)
  6. Ramsey Number R(4, 4)
  7. Geometric Application of Ramsey's Theory
  8. Coloring Points in the Plane and Elsewhere
  9. Two Colors - Two Points
  10. Three Colors - Two Points
  11. Two Colors - All Distances
  12. Two Colors on a Straight Line
  13. Two Colors - Three Points
  14. Three Colors - Bichromatic Lines
  15. Chromatic Number of the Plane
  16. Monochromatic Rectangle in a 2-coloring of the Plane
  17. Two Colors - Three Points on Circle
  18. Coloring a Graph
  19. No Equilateral Triangles, Please

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