Five teams P, Q, R, S and T played a round-robin football tournament: each team played every other team exactly once. A win earned 3 points, a draw 1 point and a loss 0 points. The following facts are known about the tournament:
1. P was the only team that did not lose any match. P and Q drew their match.
2. At the end, P had 8 points and Q had 7 points, while R, S and T finished level on points.
3. Q drew exactly one match.
4. R, S and T won exactly one match each.
5. S lost to P and to T; S's only win came against Q.
1. P was the only team that did not lose any match. P and Q drew their match.
2. At the end, P had 8 points and Q had 7 points, while R, S and T finished level on points.
3. Q drew exactly one match.
4. R, S and T won exactly one match each.
5. S lost to P and to T; S's only win came against Q.
Question 1
Numerical answer
How many matches did P win?
Show answer and solution
Answer: 2
Deduction: P (8 points, unbeaten) drew with Q and beat S, so its games against R and T gave 4 points: one win and one draw. Q (7 points) drew only with P and lost to S, so Q beat R and T. T's only win is over S, so T did not beat R. Suppose R beat S: then R–T must be a draw (neither can win again), S would have 3 points, and R and T would each have 4 plus their result against P — at least one of them draws with P, so they cannot both equal S ✗. Hence R's only win is over T, and R–S is a draw (S has no second win). Now S = 3 + 1 = 4, so R = 3 + 1 + (vs P) = 4 ⇒ R lost to P, and T = 3 + (vs P) = 4 ⇒ T drew with P.
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
P won against R and S and drew with Q and T: 2 wins (2 × 3 + 2 × 1 = 8 ✓).
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
P won against R and S and drew with Q and T: 2 wins (2 × 3 + 2 × 1 = 8 ✓).
Question 2
Numerical answer
How many matches in the tournament ended in a draw?
Show answer and solution
Answer: 3
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
Draws: P–Q, P–T and R–S. Total = 3.
Check with points: total points = 8 + 7 + 4 + 4 + 4 = 27. With 10 matches, total = 30 − (number of draws) ⇒ draws = 3 ✓ (each draw gives 2 points instead of 3).
Tip: Total points = 3 × matches − draws is a quick consistency check in 3-1-0 leagues.
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
Draws: P–Q, P–T and R–S. Total = 3.
Check with points: total points = 8 + 7 + 4 + 4 + 4 = 27. With 10 matches, total = 30 − (number of draws) ⇒ draws = 3 ✓ (each draw gives 2 points instead of 3).
Tip: Total points = 3 × matches − draws is a quick consistency check in 3-1-0 leagues.
Question 3
Single correct option
Which of the following statements is true?
- A. R beat T
- B. P beat T
- C. S beat R
- D. Q drew with R
Show answer and solution
Answer: (A) R beat T
Deduction: P (8 points, unbeaten) drew with Q and beat S, so its games against R and T gave 4 points: one win and one draw. Q (7 points) drew only with P and lost to S, so Q beat R and T. T's only win is over S, so T did not beat R. Suppose R beat S: then R–T must be a draw (neither can win again), S would have 3 points, and R and T would each have 4 plus their result against P — at least one of them draws with P, so they cannot both equal S ✗. Hence R's only win is over T, and R–S is a draw (S has no second win). Now S = 3 + 1 = 4, so R = 3 + 1 + (vs P) = 4 ⇒ R lost to P, and T = 3 + (vs P) = 4 ⇒ T drew with P.
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
R beat T is true. P drew with T (not a win); Q beat R (not a draw); R and S drew (S did not win).
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
R beat T is true. P drew with T (not a win); Q beat R (not a draw); R and S drew (S did not win).
Question 4
Single correct option
Teams level on points are ranked using only the points each earned in the matches played among those tied teams. Which team finished third in the tournament?
- A. R
- B. Q
- C. T
- D. S
Show answer and solution
Answer: (A) R
Results:
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
P (8) and Q (7) are 1st and 2nd. R, S and T are tied on 4 points. Among themselves:
- R: draw with S (1) + win over T (3) = 4
- T: win over S (3) + loss to R (0) = 3
- S: draw with R (1) + loss to T (0) = 1
So the order is R, T, S, and R finishes third (T fourth, S fifth).
| Match | Result |
|---|---|
| P vs Q | Draw |
| P vs R | P won |
| P vs S | P won |
| P vs T | Draw |
| Q vs R | Q won |
| Q vs S | S won |
| Q vs T | Q won |
| R vs S | Draw |
| R vs T | R won |
| S vs T | T won |
Points: P 8, Q 7, R 4, S 4, T 4.
P (8) and Q (7) are 1st and 2nd. R, S and T are tied on 4 points. Among themselves:
- R: draw with S (1) + win over T (3) = 4
- T: win over S (3) + loss to R (0) = 3
- S: draw with R (1) + loss to T (0) = 1
So the order is R, T, S, and R finishes third (T fourth, S fifth).
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