Boppo is a game played by 2 teams for a fixed duration (number of minutes). It is played with 8...

Phase 1: Owning the Dataset

Visual Representation

Let's create a timeline to understand the player rotation:

Game Timeline (M minutes)
|-------------------------------------------------------|
  8 on floor + 1 off floor (continuously throughout game)

Player perspective:
Each player: [---Playing (M-5 minutes)---][Rest 5 min]

Key Information:

• 8 players from each team on floor at all times
• 1 player from each team off floor at all times
• Every player rests exactly 5 minutes
• Total players per team = 8 (on floor) + 1 (off floor) = at least 9

Phase 2: Understanding the Question

We need to find values that work together:

• Number of players per team (P)
• Number of minutes per game (M)

Key Mathematical Relationship

Let's think about total rest time from two perspectives:

From game perspective:

• 1 player resting at all times for M minutes
• Total rest time available = M minutes

From players perspective:

• P players each rest 5 minutes
• Total rest time needed = P × 5 minutes

These must be equal: \(\mathrm{P} \times 5 = \mathrm{M}\)

This means:

• M must be divisible by 5
• P = M/5

Phase 3: Finding the Answer

Let's check our answer choices systematically:

Checking if choices can be number of players (P):

• If P = 9 → \(\mathrm{M} = 9 \times 5 = 45\) ✓ (45 is in our choices!)
• If P = 11 → \(\mathrm{M} = 11 \times 5 = 55\) (not in choices)
• If P = 18 → \(\mathrm{M} = 18 \times 5 = 90\) (not in choices)

Checking if choices can be game duration (M):

• If M = 30 → \(\mathrm{P} = 30/5 = 6\) (not in choices)
• If M = 45 → \(\mathrm{P} = 45/5 = 9\) ✓ (9 is in our choices!)
• If M = 50 → \(\mathrm{P} = 50/5 = 10\) (not in choices)

? Stop here - we found our answer.

Verification

With 9 players and 45-minute game:

• Each player plays: \(45 - 5 = 40\) minutes
• Each player rests: 5 minutes
• At any moment: 8 on floor, 1 off floor ✓
• Total rest across all players: \(9 \times 5 = 45\) = game duration ✓

Phase 4: Solution

Number of players per team: 9
Number of minutes per game: 45

These values satisfy all constraints: with 9 players per team and a 45-minute game, exactly 1 player can rest at all times while 8 play, and each player gets exactly 5 minutes of rest.