Problem 29 · 2021 Math Kangaroo
Stretch
Logic & Word Problems
Counting & Probability
casework
In a tournament each of the 6 teams plays one match against every other team. In each round of matches, 3 take place simultaneously. A TV station has already decided which match it will broadcast for each round, as shown in the table. In which round will team D play against team F?
| Round | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Broadcast match | A–B | C–D | A–E | E–F | A–C |
Show answer
Answer: A — 1
Show hints
Hint 1 of 2
The five rounds form a schedule where each round is three disjoint matches covering all six teams.
Still stuck? Show hint 2 →
Hint 2 of 2
Each team plays once per round, so a team's partners across the rounds are all different; reconstruct from the broadcasts.
Show solution
Approach: reconstruct the round-by-round pairing
- A's broadcast matches are B (R1), E (R3) and C (R5), so A must meet D and F in rounds 2 and 4; since R2 already shows C–D, A plays F in R2 and D in R4.
- Round 4 shows A–D and E–F, so its third match is B–C; round 2's leftover pair is B–E.
- Now D still needs B, E, F and E still needs C, D: E–C must fall in round 1 (E is busy in the others), forcing E–D into round 5.
- That leaves round 1 as A–B, C–E and the last pair D–F, so D plays F in round 1 — answer A.
Mark:
· log in to save