Four teams - Red (R), Blue (B), Green (G), and Yellow (Y) are competing in the final four rounds of the Inter-School Science Olympiad, labeled Round A, Round B, Round C, and Round D. Each round consists of one match between two teams, and every team plays exactly two matches. No team plays the same opponent more than once. The final schedule must adhere to the following rules:

• Rule 1 (Consecutive Play): The Green team (G) must play their two matches in consecutive rounds.
• Rule 2 (Fixed Appearance): The Yellow team (Y) must play in Round B.
• Rule 3 (Positional Constraint): The Red team (R) must play against the Blue team (B) in a round that is immediately before a round in which neither R nor B is playing.
• Rule 4 (Timing): The Blue team’s (B) first match must occur in an earlier round than the Green team’s (G) first match.
• Rule 5 (Opponent Link): The team that plays against the Red team (R) in the round that is not against the Blue team (B), is the same team that plays in Round D.

Identifying the Teams Excluded from a Round. Which pair of rounds contains matches where the Yellow team does not participate?

Correct Answer: Round A and Round C

To determine the rounds in which the Yellow team does not participate, let us step-by-step construct the schedule for the four rounds (Round A, Round B, Round C, and Round D) based on the given rules.

1. Understanding the Constraints:

• There are 4 rounds: Round A, Round B, Round C, and Round D.
• Each round is a match between 2 distinct teams, so 2 teams play in each round and 2 teams are excluded from that round.
• Each of the 4 teams (R, B, G, Y) plays exactly 2 matches overall.
• No two teams play each other more than once.

2. Analyzing the Rules:

• Rule 2 (Fixed Appearance): Yellow team (Y) plays in Round B.
• Rule 1 (Consecutive Play): Green team (G) plays in two consecutive rounds.
• Rule 4 (Timing): Blue team's (B) first match occurs in an earlier round than Green team's (G) first match.
• Rule 3 (Positional Constraint): Red (R) plays Blue (B) in a round that is immediately before a round in which neither R nor B plays.
• Rule 5 (Opponent Link): The team that plays R in R's other match (the match not against B) is the same team that plays in Round D.

3. Deductive Scheduling:

• Let us locate the match R vs B. By Rule 3, the round containing R vs B must be followed immediately by a round containing neither R nor B. Thus, R vs B cannot be in Round D.
• If R vs B is in Round C, then Round D has neither R nor B. But Rule 5 states that R's other opponent plays in Round D. If R vs B is in Round C, R's other opponent must play in Round A or Round B. But if Round D has neither R nor B, then Round D must consist of G vs Y. If G plays in Round D, by Rule 1 (consecutive rounds), G must also play in Round C. But Round C is R vs B, so G cannot play in Round C! This creates a contradiction.
• Therefore, R vs B cannot be in Round C.

• If R vs B is in Round A, then Round B must contain neither R nor B (by Rule 3). Thus, Round B must consist of G vs Y (since Y plays in Round B by Rule 2).
• Since B's first match is in Round A, Rule 4 (B's first match is earlier than G's first match) is satisfied because G's first match would be in Round B.
• By Rule 1, G plays consecutive rounds, so G plays in Round B and Round C.
• Now let us check R's second match. R must play one more match. Since Round B has G vs Y, R cannot play in Round B. Thus, R's second match must be in Round D.
• Who does R play in Round D? The remaining team available for Round D to pair with R. Since R's other opponent must also play in Round D (Rule 5), this is fully consistent with R playing in Round D.
• Specifically, the schedule works out seamlessly as:
- Round A: R vs B
- Round B: G vs Y
- Round C: B vs G (G's second match, B's second match)
- Round D: R vs Y (R's second match, Y's second match)

4. Verification of Rules for this Schedule:

• Round A: R vs B
• Round B: G vs Y
• Round C: B vs G
• Round D: R vs Y
- Total matches per team: R (2), B (2), G (2), Y (2). All correct.
- Rule 1: G plays in Round B and Round C (consecutive). Correct.
- Rule 2: Y plays in Round B. Correct.
- Rule 3: R vs B is in Round A, which is immediately before Round B where neither R nor B plays. Correct.
- Rule 4: B's first match is Round A, G's first match is Round B. (A is earlier than B). Correct.
- Rule 5: R's non-B opponent is Y (in Round D), who also plays in Round D. Correct.

5. Answering the Question:

• Yellow team (Y) plays in Round B and Round D.
• Therefore, Yellow team does not participate in Round A and Round C.