Daily Math Puzzle: 2026-08-20
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2026-08-20
In the annual 'Continental Clash' football tournament, 8 national teams compete. Each team plays every other team exactly once. Points are awarded as follows: 3 points for a win, 1 point for a draw, and 0 points for a loss. At the conclusion of the tournament, it was observed that the sum of all points accumulated by all 8 participating teams was exactly 75 points. Assuming no games were forfeited or canceled, how many matches in the tournament ended in a draw?
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Solution
9 — Let 'N' be the number of teams, which is 8.
First, we calculate the total number of games played in a round-robin tournament where each team plays every other team exactly once. The formula for total games (G) is N * (N - 1) / 2.
So, G = 8 * (8 - 1) / 2 = 8 * 7 / 2 = 56 / 2 = 28 games.
Next, let's consider the points awarded per game for the *sum* of both teams involved:
- If a game results in a win/loss: One team gets 3 points, the other gets 0 points. The total points contributed by that single game to the overall tournament sum is 3 + 0 = 3 points.
- If a game results in a draw: Both teams get 1 point. The total points contributed by that single game to the overall tournament sum is 1 + 1 = 2 points.
Let 'D' be the number of draws in the tournament and 'W' be the number of games that resulted in a win/loss.
We know that the total number of games G = W + D, which implies W = G - D.
The total points accumulated by all teams (given as 75) can be expressed as:
Total Points = (Points per win/loss game * Number of win/loss games) + (Points per draw game * Number of draws)
75 = (3 * W) + (2 * D)
Now, substitute W = G - D into the equation:
75 = 3 * (G - D) + 2D
75 = 3G - 3D + 2D
75 = 3G - D
Finally, substitute the value of G (total games) we calculated:
75 = 3 * 28 - D
75 = 84 - D
Solve for D:
D = 84 - 75
D = 9
Therefore, 9 matches in the tournament ended in a draw.
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