First Written: July 7, 2014

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org]

Added Sept. 14, 2026: Guoce Xin and his former grad student, Zihao Zhang, computed the cases n=7 and n=8, and independently, Gabe Vandevere.

Added Sept. 23, 2026: Read this fascinating paper by Xinru Jiang, Guoce Xin, Cheng Zhang, and Yueming Zhong. That goes way beyond.

Maple Packages

• GOALS, uses the polynomial ansatz to discover (rigorously!) polynomial expressions to lots of enumeration questions related to the Group stage of the World Cup

• WorldCup, figures out how to reconstruct the possible individual games' scores from the total score-board, and generates World-Cup puzzle books.

Some Input and Output files for the Maple package GOALS

• If you want to see polynomial expressions, in r, for the number of ways where n teams (for the World Cup take n=4) can play a round-robin tournament, where each team scored exactly r Goals For (GF) and r Goals Against (GA), in other words the number of n by n magic squares with line-sums r (famously treated in EC1, Prop. 4.6.19), BUT with the diagonal entries all 0 (teams do not play with themselves!), for n from 3 to 6,

  the input   yields the output

• If you want to see many more polynomial expressions, in r, for the number of ways where 4 teams can have Goals For, and Goals Against vectors
  (r+a1,r+a2,r+a3,r+a4)   (r+b1,r+b2,r+b3,r+b4)  
  For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥0   and   b1 ≥b2 ≥b3 ≥b4 ≥0 and a1+a2+a3+a4=b1+b2+b3+b4 ≤ 6

  the input   yields the output

• If you want to see many more polynomial expressions, in r, for the number of ways where 4 teams can have Goals For, and Goals Against vectors
  (r+a1,r+a2,r+a3,r+a4)   (r+b4,r+b3,r+b2,r+b1)  
  For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥0 and   b1 ≥b2 ≥b3 ≥b4 ≥0 and a1+a2+a3+a4=b1+b2+b3+b4 ≤ 6

  the input   yields the output

• If you want to see many more polynomial expressions, in r, for the number of ways where FIVE teams can have Goals For, and Goals Against vectors
  (r+a1,r+a2,r+a3,r+a4, r+a5)   (r+b1,r+b2,r+b3,r+b4, r+b5)  
  For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥ a5 ≥ 0 and   b1 ≥b2 ≥b3 ≥b4 ≥ b5 ≥ 0 and a1+a2+a3+a4 +a5 =b1+b2+b3+b4+b5 ≤ 4

  the input   yields the output

• If you want to see many more polynomial expressions, in r, for the number of ways where FIVE teams can have Goals For, and Goals Against vectors
  (r+a1,r+a2,r+a3,r+a4, r+a5)   (r+b5,r+b4,r+b3,r+b2, r+b1)  
  For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥ a5 ≥ 0 and   b1 ≥b2 ≥b3 ≥b4 ≥ b5 ≥ 0 and a1+a2+a3+a4 +a5 =b1+b2+b3+b4+b5 ≤ 4

  the input   yields the output

Some Input and Output files for the Maple package WorldCup

• If you want to see a Soccer puzzle book with more than seventy entertaining puzzles

  the input   yields the output

• If you want to see a Soccer puzzle book with more than twenty, somewhat more challenging puzzles

  the input   yields the output

• If you want to see a Soccer puzzle book with twelve yet more challenging puzzles

  the input   yields the output

• If you want to see a Soccer puzzle book with 22 puzzles but now there are FIVE teams

  the input   yields the output

Doron Zeilberger's Home Page