Six teams — P, Q, R, S, T and U — played a single round-robin cricket league: every team played every other team exactly once. Each match produced a winner; a win earned 2 points and a loss 0 points. There were no ties or abandoned matches.
The following facts are known about the final standings:
1. The final points of the six teams, in some order, were 8, 8, 6, 4, 2 and 2.
2. P lost exactly one match, and that was to Q.
3. Q finished with more points than R.
4. R beat Q.
5. T won exactly one match, and it was against a team that finished in the top three by points.
6. U beat T.
7. S beat every team that finished below it on points and lost to every team that finished above it.
The following facts are known about the final standings:
1. The final points of the six teams, in some order, were 8, 8, 6, 4, 2 and 2.
2. P lost exactly one match, and that was to Q.
3. Q finished with more points than R.
4. R beat Q.
5. T won exactly one match, and it was against a team that finished in the top three by points.
6. U beat T.
7. S beat every team that finished below it on points and lost to every team that finished above it.
Question 1
Single correct option
Which team did T beat?
- A. P
- B. Q
- C. R
- D. S
Show answer and solution
Answer: (C) R
T won exactly one match, against a team in the top three (points 8, 8, 6). Deduction (full results):
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
T's only win was therefore against R.
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
T's only win was therefore against R.
Question 2
Numerical answer
How many points did R finish with?
Show answer and solution
Answer: 6
From clue 2, P lost only once, so P won 4 matches (8 points). Clue 3 with clue 4: Q must also have 8 points (the only way for Q to be above R while losing to R), leaving 6 for the next team. Deduction (full results):
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
R finished with 6 points.
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
R finished with 6 points.
Question 3
Single correct option
Who won the match between Q and S?
- A. Q
- B. S
- C. Cannot be determined
- D. The match was not played
Show answer and solution
Answer: (A) Q
Deduction (full results):
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
Q's only loss was to R (Q has 4 wins out of 5), so Q beat S. Also, by clue 7, S lost to every team above it, and Q is above S.
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
Q's only loss was to R (Q has 4 wins out of 5), so Q beat S. Also, by clue 7, S lost to every team above it, and Q is above S.
Question 4
Numerical answer
How many teams beat both of the teams that finished with 8 points?
Show answer and solution
Answer: 0
Deduction (full results):
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
The 8-point teams are P and Q. The only team to beat P is Q, and the only team to beat Q is R. No team beat both, so the answer is 0.
| Team | Beat | Lost to | Points |
|---|---|---|---|
| P | R, S, T, U | Q | 8 |
| Q | P, S, T, U | R | 8 |
| R | Q, S, U | P, T | 6 |
| S | T, U | P, Q, R | 4 |
| T | R | P, Q, S, U | 2 |
| U | T | P, Q, R, S | 2 |
Key steps: P lost only to Q, so P has 4 wins (8 points). The other 8-point team also lost exactly once. It cannot be R (clue 3 puts Q above R) or T (one win); checking S or U as the second 8-pointer leaves T with no top-three team it could have beaten, so it is Q, whose only loss is to R. Next, the 6-point team: if S had 6, S would beat R, T and U, and T's lone win against a top-three team (P, Q or S) would be impossible; if U had 6, T's win would have to be over U, contradicting clue 6. So R has 6. S then has 4 and, by clue 7, beat exactly T and U. T lost to P, Q (each lost only once), S and U, so T's win was over R; U's only win was over T.
The 8-point teams are P and Q. The only team to beat P is Q, and the only team to beat Q is R. No team beat both, so the answer is 0.
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