Fitting 7 Teams Into 4 Hours a Day
The school games committee must conduct an inter-house kabaddi league for 7 houses on the single school court at the end of the term. The court is available only from 9:00 to 11:00 a.m. and from 3:00 to 5:00 p.m., that is 4 hours a day, because the academic timetable cannot be disturbed. The physical education teacher has two days. Each match needs 20 minutes of play plus 10 minutes for changeover, score sheet and warm-up, i.e., 30 minutes per slot. The committee first announces a full single round-robin league, prints the fixture, and then the teacher realises the schedule will not fit. On the first day one match also runs 8 minutes over because of an injury stoppage, and two teams finish level on league points, which starts an argument about who should be declared winner.
1. Calculate the number of matches a single round-robin league for 7 teams needs, and show with figures whether two days of 4 hours each are enough. State the shortfall.
Read the worked answer
Number of matches in a single round robin: n(n − 1)/2 = 7 × 6 / 2 = 21 matches. (Reasoning: each of the 7 teams plays the other 6, which counts every match twice, so we divide by 2.)
Slots available:
- 4 hours a day ÷ 0.5 hour per slot = 8 slots per day.
- 2 days × 8 slots = 16 slots.
Comparison: 21 matches are needed against 16 slots available, so the schedule is short by 5 slots - and that is before counting any buffer. The injury stoppage on day 1 actually consumed one extra slot, so in practice only about 14–15 matches could be completed. A printed 21-match fixture cannot be honoured, and teams would either play at the cost of class time or the league would end unfinished.
Conclusion: the round-robin format is a good choice for fairness (every house plays six matches), but it does not match the constraint. The constraint must be changed (more days, more court time) or the format must be changed - never the results.
2. Propose an alternative competition format that fits within the available slots, giving match counts, and justify why your format is fairer than simply cutting the tournament to a knockout. Include how you would seed the groups.
Read the worked answer
Recommended format: two pools, then knockout.
- Pool A: 3 teams → 3 × 2 / 2 = 3 matches.
- Pool B: 4 teams → 4 × 3 / 2 = 6 matches.
- League total = 9 matches.
- Two semifinals (A1 v B2, B1 v A2) = 2 matches.
- Third-place match + final = 2 matches.
- Total = 13 slots, out of 16, leaving 3 slots as buffer for injury stoppages, a protest or extra time.
Seeding: place last year's winner and runner-up in different pools (Pool A gets one, Pool B the other), then distribute the remaining houses by lot in the presence of house captains, and record the draw. This prevents the two strongest houses from meeting only in a dead match, and removes any suspicion of favouritism.
Why this is fairer than a plain knockout:
- A knockout needs only 6 matches, but 5 of the 7 teams would be eliminated after a single match, which defeats the purpose of an inter-house tournament in a school.
- In the pool format every team plays at least 2 matches, and the top two in each pool advance on merit over several matches, not on one lucky rally.
- The buffer of 3 slots protects the schedule from the exact problem that occurred on day 1.
Alternative if the court time can be extended: shorten each slot to 15 minutes (10 minutes play + 5 minutes changeover) so that 21 matches take 5 hours 15 minutes over two days. This is workable only if the shortened playing time is announced in advance and applied to all matches equally.
3. Two teams finish level on league points. Set out a defensible tie-break order for a school tournament and explain the reasoning behind the order. Also state how you would handle a team that is not on the court 5 minutes after its match is called.
Read the worked answer
Points system first: win = 2 points, draw = 1 point, loss = 0. This must be published before the first match, not decided after the dispute.
Tie-break order:
- Head-to-head result between the tied teams - the team that won when they met directly is placed higher. Reasoning: the most direct evidence of who is better, and it is easy for players and spectators to understand.
- Score difference across all league matches (points scored minus points conceded). Reasoning: objective, and rewards consistent performance rather than one big win; to reduce the incentive to humiliate a weak team, cap the margin counted in any one match (for example, a maximum of 10 points).
- Total points scored across the league.
- Toss of coin by the referee in the presence of both captains. Reasoning: a tie must be broken, and a public coin toss is transparent; it is the last resort, never the first.
Walkover rule: if a team is not on the court within 5 minutes of the second call, the match is awarded to the opponent with a recorded score (for example 0–5), the reason is written on the score sheet, and the defaulting captain and house master are informed in writing the same day. This prevents one team from delaying the whole schedule, which is a real constraint here with only 16 slots for 13 matches.