【Question 21】
Four football teams participate in a round-robin tournament, where each pair of teams plays one match. The winning team gets 3 points, the losing team gets 0 points, and in the case of a draw, both teams get 1 point. It is known that after all the matches, the points of the four teams are four consecutive positive integers. Find the product of these four numbers.
【Analysis and Solution】
Mathematics in sports matches.
There are a total of $C_{4}^{2}=\frac{4 \times 3}{2 \times 1}=6$ matches;
The total score of each match is 2 points or 3 points;
The minimum total score is $2 \times 6=12$ points, and the maximum total score is $3 \times 6=18$ points.
Therefore, the scores of the four teams can be $2, 3, 4, 5$ or $3, 4, 5, 6$.
If the scores of the four teams are $3, 4, 5, 6$,
the total is 18, there should be no draws, but the teams with 4 and 5 points both have draws, so this situation is discarded;
If the scores of the four teams are $2, 3, 4, 5$,
the total is 14, there should be $18-14=4$ matches that are draws,
so the team with 5 points has 2 draws, the team with 4 points has 1 draw, and the team with 2 points has 2 draws;
thus, the team with 3 points must have 3 draws;
so the team with 4 points beats the team with 2 points, and the team with 4 points loses to the team with 5 points;
the win-loss relationship of the four teams is shown in the figure:
this situation is possible;
the product of the scores of these four teams is $2 \times 3 \times 4 \times 5=120$.