Solution
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| 7 | |||
| 8 |
To solve this problem, you need to keep track of 3 important things:
- Team groups - Which teams belong to which group
- Match schedule - Which teams play each other in each round
- Games completed - Which games each team has already played
The problem states that each team plays 2 matches within the same group and 6 matches with teams from the other group.
Therefore, each team plays 2 + 6 = 8 matches total.
The total number of matches would be:
8 × 6 ÷ 2 = 24 (we divide by 2 to avoid counting duplicates, since T1 vs T3 is the same match as T3 vs T1)
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | |||
| 5 | 3 vs 6 | 2 vs 5 | |
| 6 | 1 vs 6 | ||
| 7 | |||
| 8 | 3 vs 6 | 2 vs 5 |
According to statements 3, 4, 5, and 6, this information can be directly filled in the table.
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | |||
| 5 | 3 vs 6 | 2 vs 5 | |
| 6 | 1 vs 6 | ||
| 7 | |||
| 8 | 3 vs 6 | 2 vs 5 |
According to statement 4:
Teams 1 and 5 played only once. Since the problem states that each team plays every other team in the same group only once, this means that teams 1 and 5 are in the same group.
Similarly, according to statement 3:
Teams 4 and 6 played in both round 1 and round 2. Since the problem states that each team plays each team in the other group exactly twice, this means teams 4 and 6 are in different groups.
Hence, we can conclude:
Group A: 1, 5, 6/4
Group B: 6/4
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | |||
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | ||
| 7 | |||
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
According to statement 6:
In Round 8, each team played against a team from the other group (Point 1)
Since two of the matches are (Team 3 vs Team 6) and (Team 2 vs Team 5), the remaining match should be (Team 1 vs Team 4) (according to statement 1).
Using statement 2, which says that in Round 5 and Round 8, the match-ups were identical, we can determine the matches played in rounds 5 and 8.
From this analysis, we can determine that teams 3 & 6, 2 & 5, and 1 & 4 are in different groups.
Therefore, we can now complete the group assignments as:
Group A: 1, 5, 6
Group B: 4, 2, 3
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | 4 vs 5 | ||
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | ||
| 7 | 4 vs 5 | ||
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
We know that Team 4 is in a different group from Teams 1, 5, and 6. Team 4 has already played two matches against Team 1 (in rounds 5 and 8) and two matches against Team 6 (in rounds 1 and 2).
Therefore, Team 4 must play two matches against Team 5.
These two matches can be scheduled in any two of the remaining rounds: 4, 6, or 7.
From statement 2, we know that the matches in rounds 4 and 7 are identical.
Therefore, the two matches between Team 4 and Team 5 are played in rounds 4 and 7. If it is played in round 6, then one would be in 4 or 7 which would make the total matches equal 3.
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | 4 vs 5 | ||
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | 4 vs 2 | 3 vs 5 |
| 7 | 4 vs 5 | ||
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
Group A: 1, 5, 6
Group B: 4, 2,3
Team 4 has played in all rounds except Round 6.
The only remaining team it can play against in Round 6 is Team 2.
Moreover, the only remaining match in Round 6 can be between Team 3 and Team 5.
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | ||
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | ||
| 4 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | 4 vs 2 | 3 vs 5 |
| 7 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
Group A: 1, 5, 6
Group B: 4, 2, 3
Team 2 must play two matches against Team 6. These two matches can be scheduled in rounds 3, 4, or 7.
However, if the 2 vs 6 match is played in round 3, then the other match between them would need to be played in either round 4 or 7. This would contradict statement 2, which states that the matches in rounds 4 and 7 are identical.
Therefore, the two matches between Team 2 and Team 6 must be played in rounds 4 and 7. This automatically determines that the third match in both rounds 4 and 7 is between Team 1 and Team 3.
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | 1 vs 2 | |
| 2 | 4 vs 6 | 1 vs 5 | |
| 3 | 3 vs 4 | 1 vs 2 | |
| 4 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | 4 vs 2 | 3 vs 5 |
| 7 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
Group A: 1, 5, 6
Group B: 4, 2, 3
Team 1 must play two matches against Team 2. These two matches fit perfectly in rounds 1 and 3.
| Round | Game 1 | Game 2 | Game 3 |
|---|---|---|---|
| 1 | 4 vs 6 | 1 vs 2 | 3 vs 5 |
| 2 | 4 vs 6 | 1 vs 5 | 2 vs 3 |
| 3 | 3 vs 4 | 1 vs 2 | 5 vs 6 |
| 4 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 5 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
| 6 | 1 vs 6 | 4 vs 2 | 3 vs 5 |
| 7 | 4 vs 5 | 2 vs 6 | 1 vs 3 |
| 8 | 3 vs 6 | 2 vs 5 | 1 vs 4 |
We can fill the table with remaining matches to get the final table above.
Group A: 1, 5, 6
Group B: 4, 2, 3
2,3 and 4 are in the same group while team 5 is in different group